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<!DOCTYPE html><html lang="en-US"><head><title>Annotated S4</title><meta property="og:title" content="Annotated S4"><meta charset="UTF-8"><meta name="viewport" content="width=device-width,height=device-height,initial-scale=1.0"><meta name="apple-mobile-web-app-capable" content="yes"><meta http-equiv="X-UA-Compatible" content="ie=edge"><meta property="og:type" content="website"><meta name="twitter:card" content="summary"><style>@media screen{body[data-bespoke-view=""] .bespoke-marp-parent>.bespoke-marp-osc>button,body[data-bespoke-view=next] .bespoke-marp-parent>.bespoke-marp-osc>button,body[data-bespoke-view=presenter] .bespoke-marp-presenter-container .bespoke-marp-presenter-info-container button,body[data-bespoke-view=presenter] .bespoke-marp-presenter-container .bespoke-marp-presenter-note-container button{-webkit-tap-highlight-color:transparent;-webkit-appearance:none;-moz-appearance:none;appearance:none;background-color:transparent;border:0;color:inherit;cursor:pointer;font-size:inherit;opacity:.8;outline:none;padding:0;transition:opacity .2s linear}body[data-bespoke-view=""] .bespoke-marp-parent>.bespoke-marp-osc>button:disabled,body[data-bespoke-view=next] .bespoke-marp-parent>.bespoke-marp-osc>button:disabled,body[data-bespoke-view=presenter] .bespoke-marp-presenter-container .bespoke-marp-presenter-info-container button:disabled,body[data-bespoke-view=presenter] .bespoke-marp-presenter-container .bespoke-marp-presenter-note-container button:disabled{cursor:not-allowed;opacity:.15!important}body[data-bespoke-view=""] .bespoke-marp-parent>.bespoke-marp-osc>button:hover,body[data-bespoke-view=next] .bespoke-marp-parent>.bespoke-marp-osc>button:hover,body[data-bespoke-view=presenter] .bespoke-marp-presenter-container .bespoke-marp-presenter-info-container 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<h1>Generating Extremely Long Sequences in JAX</h1>
<p><a href="http://rush-nlp.com/">Sasha Rush</a> (@srush_nlp) with <a href="https://www.siddkaramcheti.com/">Sidd Karamcheti</a></p>
<p><a href="https://github.com/srush/annotated-s4">https://github.com/srush/annotated-s4</a></p>
<p>Based on research by Albert Gu, Karan Goel, and Christopher Ré.</p>
</section>
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<h1>Intro</h1>
</section>
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<h2>Intro</h2>
<ul>
<li>
<p>Professor at Cornell / Researcher at Hugging Face</p>
</li>
<li>
<p>Open source projects</p>
<ul>
<li><a href="https://nlp.seas.harvard.edu/NamedTensor">Named Tensors</a>, <a href="https://nlp.seas.harvard.edu/2018/04/03/attention.html">Annotated Transformer</a>, <a href="https://github.com/Mini-Conf/Mini-Conf">Miniconf</a>, <a href="https://minitorch.github.io/">MiniTorch</a></li>
</ul>
</li>
</ul>
</section>
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<h2>Talk Goals</h2>
<p>Caveat: Not a research talk, there will be bugs <img class="emoji" draggable="false" alt="🧑🔬" src="https://twemoji.maxcdn.com/v/14.0.2/svg/1f9d1-200d-1f52c.svg" data-marp-twemoji=""/></p>
<ul>
<li data-marpit-fragment="1">
<ol>
<li>Learn about a new ML architecture.</li>
<li>Understand how JAX supports it.</li>
</ol>
</li>
</ul>
</section>
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<h2>JAX: Pros and Cons</h2>
<h3>Cons</h3>
<ul>
<li data-marpit-fragment="1">
<ul>
<li>Debugging is still hard</li>
<li>No NN standard</li>
<li>Hard to reason about (for me)</li>
</ul>
</li>
</ul>
<h3>Pros</h3>
<ul>
<li data-marpit-fragment="2">
<ul>
<li>Seperate math from NN (facilitates testing)</li>
<li>JIT is really impressive</li>
<li>Lifted transformations are magic</li>
</ul>
</li>
</ul>
</section>
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<h1>Problem Context</h1>
</section>
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<h2>Sequence Modeling</h2>
<p>Birds-Eye: Learning over a list of elements (discrete or sampled signal)</p>
<ul>
<li data-marpit-fragment="1">
<p>Classification</p>
<blockquote>
<p>Is the dog a good boy?</p>
<ul>
<li>Yes</li>
</ul>
</blockquote>
</li>
<li data-marpit-fragment="2">
<p>Generation</p>
<blockquote>
<p>The dog is a good _____</p>
</blockquote>
</li>
</ul>
</section>
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<h2>The Transformer</h2>
<p><img src="https://miro.medium.com/max/1400/1*YmND0Qj8O6b35J1yU_CPKQ.png" alt="" style="height:400px;" /></p>
</section>
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<h2>Transformer Dominance</h2>
<p><img src="https://pbs.twimg.com/media/FM4TD-7XIAEc2XG?format=png&name=small" alt="" style="height:400px;" /></p>
<p><a href="http://www.isattentionallyouneed.com/">isattentionallyouneed.com</a></p>
</section>
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<h2>The Transformer Weakness</h2>
<p><img src="https://miro.medium.com/max/1400/1*YmND0Qj8O6b35J1yU_CPKQ.png" alt="" style="height:300px;" /></p>
<ul>
<li>Scales <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>L</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(L^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> with length <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</li>
</ul>
</section>
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<h2>Recurrent Neural Networks (RNN)</h2>
<p><img src="https://miro.medium.com/max/1400/1*UkI9za9zTR-HL8uM15Wmzw.png" alt="" /></p>
<ul>
<li>Scales <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(L)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">L</span><span class="mclose">)</span></span></span></span> with length <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>.</li>
</ul>
</section>
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<h2>Long Range Arena</h2>
<ul>
<li>A benchmark of extremely long sequence tasks (up to 16k tokens)</li>
</ul>
<p><img src="https://d3i71xaburhd42.cloudfront.net/7e9ff94476f41041c75e253e84f487db00e9c861/6-Table1-1.png" alt="" style="height:400px;" /></p>
</section>
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<h2>Linearized Images</h2>
<p><img src="data:image/png;base64,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alt="" style="height:300px;" /></p>
</section>
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<h2>Path-X</h2>
<p><img src="https://pbs.twimg.com/media/FDTbF31UUAMgfei.png" alt="" style="height:300px;" /></p>
<ul>
<li>Classification problem on linearized (one pixel at a time) image sequence.</li>
</ul>
</section>
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<h1>Method</h1>
</section>
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<h2><a href="https://arxiv.org/abs/2111.00396">Efficiently Modeling Long Sequences with Structured State Spaces</a></h2>
<p>Albert Gu, Karan Goel, and Christopher Ré.</p>
<p><img src="images/hero.png" alt="" /></p>
</section>
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<h2>Punchline</h2>
<p><img src="images/table.png" alt="" /></p>
</section>
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<h1>Challenges</h1>
<ul>
<li>The model is quite mathematically complicated (want to test)</li>
<li>Core operations required external libraries in Torch</li>
<li>Follow-up work uses similar structure</li>
</ul>
<h1>Goal</h1>
<ul>
<li data-marpit-fragment="1">A concise pedagogical JAX / Flax implementation.</li>
</ul>
</section>
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<h1><a href="https://srush.github.io/annotated-s4/">The Annotated S4</a></h1>
</section>
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<h2>Image Generation</h2>
<p><img src="images/im0.4.png" alt="" /></p>
</section>
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<h2>Speech Generation</h2>
<p><img src="images/speech25.0.png" alt="" style="height:500px;" /></p>
</section>
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<h1>Part 1: SSM</h1>
</section>
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<h2>State Space Models (SSM)</h2>
<ul>
<li>A <a href="https://en.wikipedia.org/wiki/State-space_representation">state space model</a> maps a 1-D input signal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> to an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>-D latent state <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><br />
before projecting to a 1-D output signal <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</li>
</ul>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">A</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="bold-italic">B</mi><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">C</mi><mi>x</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned}
x'(t) &= \boldsymbol{A}x(t) + \boldsymbol{B}u(t) \\
y(t) &= \boldsymbol{C}x(t)\\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">A</span></span></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">A</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">A</span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">B</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">C</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span></span></span></span> are parameters; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> input, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> output, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> state</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">random_SSM</span>(<span class="hljs-params">rng, N</span>):</span>
a_r, b_r, c_r = jax.random.split(rng, <span class="hljs-number">3</span>)
A = jax.random.uniform(a_r, (N, N))
B = jax.random.uniform(b_r, (N, <span class="hljs-number">1</span>))
C = jax.random.uniform(c_r, (<span class="hljs-number">1</span>, N))
<span class="hljs-keyword">return</span> A, B, C
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="24" data-paginate="true" data-theme="default" data-marpit-pagination="24" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Discretization</h2>
<ul>
<li>
<p>To discretize input sequence <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>u</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><msub><mi>u</mi><mrow><mi>L</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u_0, u_1, \dots, u_{L-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> need a <strong>step size</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span> representing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>k</mi></msub><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>k</mi><mi mathvariant="normal">Δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u_k = u(k \Delta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord">Δ</span><span class="mclose">)</span></span></span></span>.</p>
</li>
<li>
<p>One choice for discretization is a <a href="https://en.wikipedia.org/wiki/Bilinear_transform">bilinear transform</a>.</p>
</li>
</ul>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>−</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">/</mi><mn>2</mn><mo>⋅</mo><mi mathvariant="bold-italic">A</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">/</mi><mn>2</mn><mo>⋅</mo><mi mathvariant="bold-italic">A</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>−</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">/</mi><mn>2</mn><mo>⋅</mo><mi mathvariant="bold-italic">A</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">B</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="bold-italic">C</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\boldsymbol{\overline{A}} &= (\boldsymbol{I} - \Delta/2 \cdot \boldsymbol{A})^{-1}(\boldsymbol{I} + \Delta/2 \cdot \boldsymbol{A}) \\
\boldsymbol{\overline{B}} &= (\boldsymbol{I} - \Delta/2 \cdot \boldsymbol{A})^{-1} \Delta \boldsymbol{B} \\
\boldsymbol{\overline{C}} &= \boldsymbol{C}\\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6383em;vertical-align:-2.0692em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5692em;"><span style="top:-4.6831em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></span><span style="top:-3.1369em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></span><span style="top:-1.5908em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.0692em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5692em;"><span style="top:-4.6831em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">Δ/2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">A</span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">Δ/2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">A</span></span></span><span class="mclose">)</span></span></span><span style="top:-3.1369em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">Δ/2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol">A</span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">Δ</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span></span></span><span style="top:-1.5908em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.0692em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">discretize</span>(<span class="hljs-params">A, B, C, step</span>):</span>
I = np.eye(A.shape[<span class="hljs-number">0</span>])
BL = inv(I - (step / <span class="hljs-number">2.0</span>) * A)
Ab = BL @ (I + (step / <span class="hljs-number">2.0</span>) * A)
Bb = (BL * step) @ B
<span class="hljs-keyword">return</span> Ab, Bb, C
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="25" data-paginate="true" data-theme="default" data-marpit-pagination="25" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Discretized SSM as RNN</h2>
<ul>
<li>Once discretized with step <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Δ</span></span></span></span>, the SSM can be viewed as a linear RNN,</li>
</ul>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>y</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
x_{k} &= \boldsymbol{\overline{A}} x_{k-1} + \boldsymbol{\overline{B}} u_k\\
y_k &= \boldsymbol{\overline{C}} x_k \\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0922em;vertical-align:-1.2961em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7961em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3639em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2961em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7961em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3639em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2961em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">scan_SSM</span>(<span class="hljs-params">Ab, Bb, Cb, u, x0</span>):</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">step</span>(<span class="hljs-params">x_k_1, u_k</span>):</span>
x_k = Ab @ x_k_1 + Bb @ u_k
y_k = Cb @ x_k
<span class="hljs-keyword">return</span> x_k, y_k
<span class="hljs-keyword">return</span> jax.lax.scan(step, x0, u)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="26" data-paginate="true" data-theme="default" data-marpit-pagination="26" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Tangent: A Mechanics Example</h2>
<ul>
<li>
<p><a href="https://en.wikipedia.org/wiki/State-space_representation#Moving_object_example">Example from mechanics</a>, mass on a spring</p>
<ul>
<li>forward position <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></li>
<li>force <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> is applied to this mass</li>
<li>parameterized by mass (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>), spring constant (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>), friction constant (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>)</li>
</ul>
</li>
</ul>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>m</mi><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>b</mi><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
my''(t) = u(t) - by'(t) - ky(t)
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5em;vertical-align:-0.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1em;"><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p></section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="27" data-paginate="true" data-theme="default" data-marpit-pagination="27" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Tangent: A Mechanics Example [Matrix form]</h2>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>m</mi><msup><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>b</mi><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
my''(t) = u(t) - by'(t) - ky(t)
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5em;vertical-align:-0.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1em;"><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">u</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right" columnspacing="0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">A</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>k</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>b</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi mathvariant="bold-italic">B</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mi>m</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="bold-italic">C</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\boldsymbol{A} &= \begin{bmatrix} 0 & 1 \\ -k/m & -b/m \end{bmatrix} \\
\boldsymbol{B} & = \begin{bmatrix} 0 \\ 1/m \end{bmatrix} & \boldsymbol{C} = \begin{bmatrix} 1 & 0 \end{bmatrix} \\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.4001em;vertical-align:-2.45em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol">A</span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord">/</span><span class="mord mathnormal">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">b</span><span class="mord">/</span><span class="mord mathnormal">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1/</span><span class="mord mathnormal">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span style="top:-2.25em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">example_mass</span>(<span class="hljs-params">k, b, m</span>):</span>
A = np.array([[<span class="hljs-number">0</span>, <span class="hljs-number">1</span>], [-k / m, -b / m]])
B = np.array([[<span class="hljs-number">0</span>], [<span class="hljs-number">1.0</span> / m]])
C = np.array([[<span class="hljs-number">1.0</span>, <span class="hljs-number">0</span>]])
<span class="hljs-keyword">return</span> A, B, C
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="28" data-paginate="true" data-theme="default" data-marpit-pagination="28" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Tangent: A Mechanics Example (with force)</h2>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-meta">@partial(<span class="hljs-params">np.vectorize, signature=<span class="hljs-string">"()->()"</span></span>)</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">example_force</span>(<span class="hljs-params">t</span>):</span>
x = np.sin(<span class="hljs-number">10</span> * t)
<span class="hljs-keyword">return</span> x * (x > <span class="hljs-number">0.5</span>)
</span></span></foreignObject></svg></code></pre>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">example_ssm</span>(<span class="hljs-params">L=<span class="hljs-number">100</span></span>):</span>
ssm = example_mass(k=<span class="hljs-number">40</span>, b=<span class="hljs-number">5</span>, m=<span class="hljs-number">1</span>)
<span class="hljs-comment"># L samples of u(t).</span>
step = <span class="hljs-number">1.0</span> / L
ks = np.arange(L)
u = example_force(ks * step)
y = scan_SSM(*ssm, u)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="29" data-paginate="true" data-theme="default" data-marpit-pagination="29" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="line.gif" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="30" data-paginate="true" data-theme="default" data-marpit-pagination="30" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Training SSMs</h2>
<ul>
<li>Our Goal: Train a neural network with SSMs</li>
<li>SSM RNNs: Fast for generation, but slow for training</li>
</ul>
<h3>Key Properties</h3>
<ul>
<li>SSM CNNs: Slow for generation, but fast for training</li>
<li>Initilization</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="31" data-paginate="true" data-theme="default" data-marpit-pagination="31" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>SSMs as wide CNNs</h2>
<ol>
<li>"Unroll" the RNN representation</li>
</ol>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>y</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
x_{k} &= \boldsymbol{\overline{A}} x_{k-1} + \boldsymbol{\overline{B}} u_k\\
y_k &= \boldsymbol{\overline{C}} x_k \\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0922em;vertical-align:-1.2961em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7961em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3639em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2961em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7961em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3639em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2961em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right left right left right" columnspacing="0em 1em 0em 1em 0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>x</mi><mn>0</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>1</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mn>2</mn></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>y</mi><mn>0</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>y</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>1</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>y</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mn>2</mn></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
x_0 &= \boldsymbol{\overline{B}} u_0 &
x_1 &= \boldsymbol{\overline{A}} \boldsymbol{\overline{B}} u_0 + \boldsymbol{\overline{B}} u_1 &
x_2 &= \boldsymbol{\overline{A}}^2 \boldsymbol{\overline{B}} u_0 + \boldsymbol{\overline{A}} \boldsymbol{\overline{B}} u_1 + \boldsymbol{\overline{B}} u_2 & \dots
\\
y_0 &= \boldsymbol{\overline{C}} \boldsymbol{\overline{B}} u_0 &
y_1 &= \boldsymbol{\overline{C}} \boldsymbol{\overline{A}} \boldsymbol{\overline{B}} u_0 + \boldsymbol{\overline{C}} \boldsymbol{\overline{B}} u_1 &
y_2 &= \boldsymbol{\overline{C}} \boldsymbol{\overline{A}}^2 \boldsymbol{\overline{B}} u_0 + \boldsymbol{\overline{C}} \boldsymbol{\overline{A}} \boldsymbol{\overline{B}} u_1 + \boldsymbol{\overline{C}} \boldsymbol{\overline{B}} u_2
& \dots
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.5002em;vertical-align:-1.5001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0901em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0901em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0001em;"><span style="top:-4.0001em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.0901em;"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5001em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p></section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="32" data-paginate="true" data-theme="default" data-marpit-pagination="32" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>SSMs as wide CNNs</h2>
<ol start="2">
<li>Form a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>-length kernel</li>
</ol>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>y</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi>k</mi></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>0</mn></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msub><mi>u</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
y_k &= \boldsymbol{\overline{C}} \boldsymbol{\overline{A}}^k \boldsymbol{\overline{B}} u_0 + \boldsymbol{\overline{C}} \boldsymbol{\overline{A}}^{k-1} \boldsymbol{\overline{B}} u_1 + \dots + \boldsymbol{\overline{C}} \boldsymbol{\overline{A}} \boldsymbol{\overline{B}} u_{k-1} + \boldsymbol{\overline{C}}\boldsymbol{\overline{B}} u_k
\\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7851em;vertical-align:-0.6426em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1426em;"><span style="top:-3.1426em;"><span class="pstrut" style="height:3.1251em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6426em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1426em;"><span style="top:-3.1426em;"><span class="pstrut" style="height:3.1251em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1251em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1251em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6426em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><blockquote></blockquote>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi><mover accent="true"><mi mathvariant="bold-italic">K</mi><mo stretchy="true">‾</mo></mover></mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mi>L</mi></msup><mo>=</mo><mo stretchy="false">(</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><mo separator="true">,</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mrow><mi>L</mi><mo>−</mo><mn>1</mn></mrow></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\boldsymbol{\overline{K}} \in \mathbb{R}^L = (\boldsymbol{\overline{C}}\boldsymbol{\overline{B}}, \boldsymbol{\overline{C}}\boldsymbol{\overline{A}}\boldsymbol{\overline{B}}, \dots, \boldsymbol{\overline{C}}\boldsymbol{\overline{A}}^{L-1}\boldsymbol{\overline{B}})
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7773em;vertical-align:-0.6387em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1387em;"><span style="top:-3.1387em;"><span class="pstrut" style="height:3.1173em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">K</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathbb">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1173em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.6387em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">K_conv</span>(<span class="hljs-params">Ab, Bb, Cb, L</span>):</span>
<span class="hljs-keyword">return</span> np.array(
[(Cb @ matrix_power(Ab, l) @ Bb).reshape() <span class="hljs-keyword">for</span> l <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(L)]
)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="33" data-paginate="true" data-theme="default" data-marpit-pagination="33" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>SSMs as wide CNNs</h2>
<ol start="3">
<li>Apply as a (non-cicular) convolution</li>
</ol>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">K</mi><mo stretchy="true">‾</mo></mover></mi><mo>∗</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">y = \boldsymbol{\overline{K}} \ast u
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8861em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">K</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">non_circular_convolution</span>(<span class="hljs-params">u, K, nofft=<span class="hljs-literal">False</span></span>):</span>
<span class="hljs-keyword">if</span> nofft:
<span class="hljs-keyword">return</span> convolve(u, K, mode=<span class="hljs-string">"full"</span>)[: u.shape[<span class="hljs-number">0</span>]]
<span class="hljs-keyword">else</span>:
ud = np.fft.rfft(np.pad(u, (<span class="hljs-number">0</span>, K.shape[<span class="hljs-number">0</span>])))
Kd = np.fft.rfft(np.pad(K, (<span class="hljs-number">0</span>, u.shape[<span class="hljs-number">0</span>])))
<span class="hljs-keyword">return</span> np.fft.irfft(ud * Kd)[: u.shape[<span class="hljs-number">0</span>]]
</span></span></foreignObject></svg></code></pre>
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>L</mi><mi>log</mi><mo></mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(L \log L)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">L</span><span class="mclose">)</span></span></span></span> training through FFT</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="34" data-paginate="true" data-theme="default" data-marpit-pagination="34" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Initialization with <a href="https://arxiv.org/abs/2008.07669">HiPPO</a></h2>
<ul>
<li>Fast training, but random init does terribly. MNIST classification benchmark <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">50\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">50%</span></span></span></span>.</li>
<li>HiPPO initialization of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\mathbf{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">A</span></span></span></span> improves this number to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>98</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">98\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">98%</span></span></span></span></li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">make_HiPPO</span>(<span class="hljs-params">N</span>):</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">v</span>(<span class="hljs-params">n, k</span>):</span>
<span class="hljs-keyword">if</span> n > k:
<span class="hljs-keyword">return</span> np.sqrt(<span class="hljs-number">2</span> * n + <span class="hljs-number">1</span>) * np.sqrt(<span class="hljs-number">2</span> * k + <span class="hljs-number">1</span>)
<span class="hljs-keyword">elif</span> n == k:
<span class="hljs-keyword">return</span> n + <span class="hljs-number">1</span>
<span class="hljs-keyword">else</span>:
<span class="hljs-keyword">return</span> <span class="hljs-number">0</span>
mat = [[v(n, k) <span class="hljs-keyword">for</span> k <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, N + <span class="hljs-number">1</span>)] <span class="hljs-keyword">for</span> n <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, N + <span class="hljs-number">1</span>)]
<span class="hljs-keyword">return</span> -np.array(mat)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="35" data-paginate="true" data-theme="default" data-marpit-pagination="35" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>HiPPO Intuition Sketch</h2>
<ul>
<li>Recall <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span>-dimensional hidden representation of an <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>-step signal</li>
<li>HiPPO approximates state as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> Legendre coefficients representing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>.</li>
</ul>
<p><img src="images/leg.png" alt="" style="height:300px;" /></p>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">example_legendre</span>(<span class="hljs-params">N=<span class="hljs-number">8</span></span>):</span>
u = (np.random.rand(N) - <span class="hljs-number">0.5</span>) * <span class="hljs-number">2</span>
t = np.linspace(-<span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">100</span>)
x = numpy.polynomial.legendre.Legendre(u)(t)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="36" data-paginate="true" data-theme="default" data-marpit-pagination="36" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Tangent: Neat JAX things.</h2>
<ul>
<li>Everything is a modular testable function</li>
<li>So far - no parameter, batches, NN nonsense</li>
<li>In fact, mostly scalar modeling.</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="37" data-paginate="true" data-theme="default" data-marpit-pagination="37" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>SSM Network Layer</h2>
<ul>
<li>SSM layer with Flax (still scalar!)</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">SSMLayer</span>(<span class="hljs-params">nn.Module</span>):</span>
A: np.DeviceArray <span class="hljs-comment"># HiPPO</span>
N, L: <span class="hljs-built_in">int</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">setup</span>(<span class="hljs-params">self</span>):</span>
self.B = self.param(<span class="hljs-string">"B"</span>, lecun_normal(), (self.N, <span class="hljs-number">1</span>))
self.C = self.param(<span class="hljs-string">"C"</span>, lecun_normal(), (<span class="hljs-number">1</span>, self.N))
self.step = np.exp(self.param(<span class="hljs-string">"log_step"</span>, log_step_initializer(), (<span class="hljs-number">1</span>,)))
<span class="hljs-comment"># Conv created each time during training</span>
self.ssm = discretize(self.A, self.B, self.C, step=self.step)
self.K = K_conv(*self.ssm, self.L)
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">__call__</span>(<span class="hljs-params">self, u</span>):</span>
<span class="hljs-keyword">return</span> non_circular_convolution(u, self.K)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="38" data-marpit-fragments="1" data-paginate="true" data-theme="default" data-marpit-pagination="38" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Lifting SSM Layer</h2>
<ul>
<li>Lift to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> copies</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap>nn.vmap(
layer, in_axes=<span class="hljs-number">1</span>, out_axes=<span class="hljs-number">1</span>,
variable_axes={<span class="hljs-string">"params"</span>: <span class="hljs-number">1</span>}, <span class="hljs-comment"># New Params</span>
split_rngs={<span class="hljs-string">"params"</span>: <span class="hljs-literal">True</span>},
)
</span></span></foreignObject></svg></code></pre>
<ul>
<li>Over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> batches</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap>nn.vmap(
layer, in_axes=<span class="hljs-number">0</span>, out_axes=<span class="hljs-number">0</span>,
variable_axes={<span class="hljs-string">"params"</span>: <span class="hljs-literal">None</span>}, <span class="hljs-comment"># Shared Params</span>
split_rngs={<span class="hljs-string">"params"</span>: <span class="hljs-literal">False</span>},
)
</span></span></foreignObject></svg></code></pre>
<ul>
<li data-marpit-fragment="1">Put into a stack of layers (similar to Transformers)</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="39" data-paginate="true" data-theme="default" data-marpit-pagination="39" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>SSM RNN Layer</h2>
<ul>
<li>Alternative SSM layer with Flax Caching</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">SSMRNNLayer</span>(<span class="hljs-params">nn.Module</span>):</span>
A: np.DeviceArray <span class="hljs-comment"># HiPPO</span>
N, L: <span class="hljs-built_in">int</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">setup</span>(<span class="hljs-params">self</span>):</span>
self.B = self.param(<span class="hljs-string">"B"</span>, lecun_normal(), (self.N, <span class="hljs-number">1</span>))
self.C = self.param(<span class="hljs-string">"C"</span>, lecun_normal(), (<span class="hljs-number">1</span>, self.N))
self.step = np.exp(self.param(<span class="hljs-string">"log_step"</span>, log_step_initializer(), (<span class="hljs-number">1</span>,)))
self.ssm = discretize(self.A, self.B, self.C, step=self.step)
self.x_k_1 = self.variable(<span class="hljs-string">"cache"</span>, <span class="hljs-string">"cache_x_k"</span>, np.zeros, (self.N,))
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">__call__</span>(<span class="hljs-params">self, u</span>):</span>
x_k, y_s = scan_SSM(*self.ssm, u[:, np.newaxis], self.x_k_1.value)
<span class="hljs-keyword">if</span> self.is_mutable_collection(<span class="hljs-string">"cache"</span>):
self.x_k_1.value = x_k
<span class="hljs-keyword">return</span> y_s.reshape(-<span class="hljs-number">1</span>).real + self.D * u
</span></span></foreignObject></svg></code></pre>
</section>
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<h1>Part 2: S4</h1>
</section>
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<h2>Issue: Calculating <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span></h2>
<ul>
<li>Unfortunately, this step is a problem.</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">K_conv</span>(<span class="hljs-params">Ab, Bb, Cb, L</span>):</span>
<span class="hljs-keyword">return</span> np.array(
[(Cb @ matrix_power(Ab, l) @ Bb).reshape() <span class="hljs-keyword">for</span> l <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(L)]
)
</span></span></foreignObject></svg></code></pre>
<ul>
<li>
<p>Main contribution of S4 is to fix this function.</p>
</li>
<li>
<p>Today: quick sketch of how it works</p>
</li>
</ul>
</section>
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<h2>Two S4 Tricks</h2>
<p>See blog post for full details. Here are two neat JAX tricks.</p>
<ul>
<li>
<p>Instead of computing <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi><mover accent="true"><mi mathvariant="bold-italic">K</mi><mo stretchy="true">‾</mo></mover></mi></mrow><annotation encoding="application/x-tex">\boldsymbol{\overline{K}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8861em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">K</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></span></span> directly, S4 evaluates its <strong><a href="https://en.wikipedia.org/wiki/Generating_function">truncated generating function</a></strong>.</p>
<ul>
<li>This becomes a functional <code>vmap</code> in JAX.</li>
</ul>
</li>
<li>
<p>In order to evalute the generating function it computes a <strong><a href="https://en.wikipedia.org/wiki/Cauchy_matrix">Cauchy kernel</a></strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mrow><msub><mi>ω</mi><mi>j</mi></msub><mo>−</mo><msub><mi>ζ</mi><mi>k</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{\omega_j - \zeta_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3874em;vertical-align:-0.5423em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07378em;">ζ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.0738em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5423em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p>
<ul>
<li>This is intractable in Torch, but is jitted out in JAX.</li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="43" data-paginate="true" data-theme="default" data-marpit-pagination="43" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h3>Trick 1. SSM Generating Functions</h3>
<p>The <em>truncated SSM generating function</em> at node <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> with truncation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> is</p>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi mathvariant="script">K</mi><mo>^</mo></mover><mi>L</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo separator="true">;</mo><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mo separator="true">,</mo><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><mo separator="true">,</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>L</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi>i</mi></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi>z</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\hat{\mathcal{K}}_L(z; \boldsymbol{\overline{A}}, \boldsymbol{\overline{B}}, \boldsymbol{\overline{C}}) \in \mathbb{C} := \sum_{i=0}^{L-1} \boldsymbol{\overline{C}} \boldsymbol{\overline{A}}^i \boldsymbol{\overline{B}} z^i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathcal" style="margin-right:0.01445em;">K</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6889em;"></span><span class="mord mathbb">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1007em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">K_gen_naive</span>(<span class="hljs-params">Ab, Bb, Cb, L</span>):</span>
K = K_conv(Ab, Bb, Cb, L)
<span class="hljs-keyword">return</span> <span class="hljs-keyword">lambda</span> z: np.<span class="hljs-built_in">sum</span>(K * (z ** np.arange(L)))
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="44" data-paginate="true" data-theme="default" data-marpit-pagination="44" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h3>Trick 1. SSM Generating Functions</h3>
<p>We can recover the kernel <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">K</mi></mrow><annotation encoding="application/x-tex">{\cal K}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord"><span class="mord mathcal" style="margin-right:0.01445em;">K</span></span></span></span></span></span> through a <a href="https://en.wikipedia.org/wiki/Z-transform#Inverse_Z-transform">z-transform at the roots of unity</a><br />
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mo stretchy="false">{</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mfrac><mi>k</mi><mi>L</mi></mfrac><mo>:</mo><mi>k</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>L</mi><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Omega = \{ \exp(2\pi \frac{k}{L} : k \in [L] \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2251em;vertical-align:-0.345em;"></span><span class="mopen">{</span><span class="mop">exp</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8801em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">L</span><span class="mclose">]}</span></span></span></span> and inverse fourier transformation.</p>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">conv_from_gen</span>(<span class="hljs-params">gen, L</span>):</span>
Omega_L = np.exp((-<span class="hljs-number">2j</span> * np.pi) * (np.arange(L) / L))
atRoots = jax.vmap(gen)(Omega_L)
<span class="hljs-keyword">return</span> np.fft.ifft(atRoots, L).reshape(L).real
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="45" data-paginate="true" data-theme="default" data-marpit-pagination="45" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h3>Trick 1. SSM Generating Functions</h3>
<p>Simplifying the generating function allows us to avoid calling <code>K_conv</code></p>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi mathvariant="script">K</mi><mo>^</mo></mover><mi>L</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>L</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi>i</mi></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi><msup><mi>z</mi><mi>i</mi></msup><mo>=</mo><mi><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo stretchy="true">‾</mo></mover></mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>−</mo><msup><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi>L</mi></msup><msup><mi>z</mi><mi>L</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">I</mi><mo>−</mo><mi><mover accent="true"><mi mathvariant="bold-italic">A</mi><mo stretchy="true">‾</mo></mover></mi><mi>z</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo stretchy="true">‾</mo></mover></mi></mrow><annotation encoding="application/x-tex">\hat{\mathcal{K}}_L(z) = \sum_{i=0}^{L-1} \boldsymbol{\overline{C}} \boldsymbol{\overline{A}}^i \boldsymbol{\overline{B}} z^i = \boldsymbol{\overline{C}} (\boldsymbol{I} - \boldsymbol{\overline{A}}^L z^L) (\boldsymbol{I} - \boldsymbol{\overline{A}} z)^{-1} \boldsymbol{\overline{B}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1968em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathcal" style="margin-right:0.01445em;">K</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0144em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">L</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1007em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1361em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.06979em;">C</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.3673em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1173em;"><span style="top:-3.339em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord boldsymbol" style="margin-right:0.07778em;">I</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1361em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol">A</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8861em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord boldsymbol" style="margin-right:0.04835em;">B</span></span></span><span style="top:-3.8061em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">K_gen_inverse</span>(<span class="hljs-params">Ab, Bb, Cb, L</span>):</span>
I = np.eye(Ab.shape[<span class="hljs-number">0</span>])
Ab_L = matrix_power(Ab, L)
Ct = Cb @ (I - Ab_L)
<span class="hljs-keyword">return</span> <span class="hljs-keyword">lambda</span> z: (Ct.conj() @ inv(I - Ab * z) @ Bb).reshape()
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="46" data-paginate="true" data-theme="default" data-marpit-pagination="46" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h3>Trick 2. Exploiting Structure</h3>
<p>Under a diagonal assumption on <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi><mo>=</mo><mi mathvariant="normal">Λ</mi></mrow><annotation encoding="application/x-tex">\mathbf{A}=\Lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Λ</span></span></span></span> you can further reduce the generating function to the following kernel form,</p>
<p><svg data-marp-fitting="svg" data-marp-fitting-math><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi><mover accent="true"><mi mathvariant="bold-italic">K</mi><mo>^</mo></mover></mi><mi mathvariant="bold">Λ</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>c</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><munder><mo>∑</mo><mi>i</mi></munder><mfrac><mrow><msub><mover accent="true"><mi>C</mi><mo>~</mo></mover><mi>i</mi></msub><msub><mi>B</mi><mi>i</mi></msub></mrow><mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi mathvariant="normal">Λ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned}
\boldsymbol{\hat{K}}_{\boldsymbol{\Lambda}}(z) & = c(z) \sum_i \frac{\tilde{C}_i B_i} {(g(z) - \Lambda_{i})} \\
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.1749em;vertical-align:-1.3374em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8374em;"><span style="top:-3.8374em;"><span class="pstrut" style="height:3.5972em;"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9495em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord boldsymbol" style="margin-right:0.06979em;">K</span></span><span style="top:-3.2551em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2875em;"><span class="mord mathbf">^</span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3303em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">Λ</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3374em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8374em;"><span style="top:-3.8374em;"><span class="pstrut" style="height:3.5972em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">c</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5972em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord">Λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.3374em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></foreignObject></svg></p><p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span> is a constant, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span> is a function of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>.</p>
<ul>
<li>
<p>However the transform of this function is memory and compute-intensive.</p>
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo><mn>16</mn><mo separator="true">,</mo><mn>000</mn></mrow><annotation encoding="application/x-tex">L=16,000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">16</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">000</span></span></span></span> different <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span></span> different <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span></li>
<li>Instantiating full tensor is intractable</li>
<li>Libraries like KeOps avoid this issue</li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="47" data-paginate="true" data-theme="default" data-marpit-pagination="47" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h3>Trick 2. Exploiting Structure</h3>
<p>In JAX we can rely on the JIT to take care of this for us.</p>
<ul>
<li>JIT handles the fusion of the sum term</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-meta">@partial(<span class="hljs-params">np.vectorize, signature=<span class="hljs-string">"(c),(),(c)->()"</span></span>)</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">cauchy_dot</span>(<span class="hljs-params">v, omega, lambd</span>):</span>
<span class="hljs-keyword">return</span> (v / (omega - lambd)).<span class="hljs-built_in">sum</span>()
</span></span></foreignObject></svg></code></pre>
<ul>
<li>JAX <code>remat</code> handles cases of very long sequences.</li>
</ul>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap>jax.remat(cauchy_dot)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="48" data-paginate="true" data-theme="default" data-marpit-pagination="48" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h1>Part 3: S4 in Practice</h1>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="49" data-marpit-fragments="1" data-paginate="true" data-theme="default" data-marpit-pagination="49" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Training S4</h2>
<ul>
<li>So far: tested code for training S4 as a CNN and running it as an RNN.</li>
</ul>
<ul>
<li data-marpit-fragment="1">MNIST classification and CIFAR classification (by pixel) are strong.</li>
</ul>
<p><img src="https://user-images.githubusercontent.com/3983652/157948685-5b836f87-7db3-42d7-8415-7cc544c97ce4.png" alt="" style="height:400px;" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="50" data-paginate="true" data-theme="default" data-marpit-pagination="50" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Goal</h2>
<ul>
<li>Generate extremely long sequences.</li>
</ul>
<p><img src="https://miro.medium.com/max/1400/1*UkI9za9zTR-HL8uM15Wmzw.png" alt="" /></p>
<ul>
<li>Expreriments on MNIST, QuickDraw, SpeechCommands</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="51" data-paginate="true" data-theme="default" data-marpit-pagination="51" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>S4 Model</h2>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="52" data-paginate="true" data-theme="default" data-marpit-pagination="52" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Training to Generate by Pixeal</h2>
<p>Code to sample from the RNN</p>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">sample</span>(<span class="hljs-params">model, params, prime, cache, x, start, end, rng</span>):</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">loop</span>(<span class="hljs-params">i, cur</span>):</span>
x, rng, cache = cur
r, rng = jax.random.split(rng)
out, <span class="hljs-built_in">vars</span> = model.apply(
{<span class="hljs-string">"params"</span>: params, <span class="hljs-string">"cache"</span>: cache},
x[:, np.arange(<span class="hljs-number">1</span>, <span class="hljs-number">2</span>) * i],
mutable=[<span class="hljs-string">"cache"</span>],
)
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">update</span>(<span class="hljs-params">x, out</span>):</span>
p = jax.random.categorical(r, out[<span class="hljs-number">0</span>])
<span class="hljs-keyword">return</span> x.at[i + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>].<span class="hljs-built_in">set</span>(p)
x = jax.vmap(update)(x, out)
<span class="hljs-keyword">return</span> x, rng, <span class="hljs-built_in">vars</span>[<span class="hljs-string">"cache"</span>].unfreeze()
<span class="hljs-keyword">return</span> jax.lax.fori_loop(start, end, loop, (x, rng, cache))[<span class="hljs-number">0</span>]
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="53" data-paginate="true" data-theme="default" data-marpit-pagination="53" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Generating by Pixel</h2>
<p><img src="images/sample.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="54" data-paginate="true" data-theme="default" data-marpit-pagination="54" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Prefix Generation</h2>
<p><img src="images/im0.1.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="55" data-paginate="true" data-theme="default" data-marpit-pagination="55" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/im0.2.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="56" data-paginate="true" data-theme="default" data-marpit-pagination="56" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/im0.3.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="57" data-paginate="true" data-theme="default" data-marpit-pagination="57" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/im0.4.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="58" data-paginate="true" data-theme="default" data-marpit-pagination="58" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/im0.5.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="59" data-paginate="true" data-theme="default" data-marpit-pagination="59" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Experiments: QuickDraw</h2>
<p><img src="images/quickdraw/im1.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="60" data-paginate="true" data-theme="default" data-marpit-pagination="60" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/quickdraw/im2.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="61" data-paginate="true" data-theme="default" data-marpit-pagination="61" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/quickdraw/im3.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="62" data-paginate="true" data-theme="default" data-marpit-pagination="62" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/quickdraw/im4.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="63" data-paginate="true" data-theme="default" data-marpit-pagination="63" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Experiments: Sound</h2>
<p><img src="images/speech3.1.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="64" data-paginate="true" data-theme="default" data-marpit-pagination="64" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/speech6.1.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="65" data-paginate="true" data-theme="default" data-marpit-pagination="65" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/speech7.0.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="66" data-paginate="true" data-theme="default" data-marpit-pagination="66" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/speech9.0.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="67" data-paginate="true" data-theme="default" data-marpit-pagination="67" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/speech10.0.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="68" data-paginate="true" data-theme="default" data-marpit-pagination="68" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<p><img src="images/speech13.1.png" alt="" /></p>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="69" data-paginate="true" data-theme="default" data-marpit-pagination="69" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h1>Conclusion & Future Work</h1>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="70" data-paginate="true" data-theme="default" data-marpit-pagination="70" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Conclusion (on JAX)</h2>
<ul>
<li>
<p>JAX really signs at modular mathematical code.</p>
</li>
<li>
<p>JAX JIT makes some hard code trivial.</p>
</li>
<li>
<p>Lifting in Flax</p>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="71" data-paginate="true" data-theme="default" data-marpit-pagination="71" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>New Paper - <a href="https://arxiv.org/abs/2203.14343">Diagonal State Spaces</a>.</h2>
<pre><code class="language-python"><svg data-marp-fitting="svg" data-marp-fitting-code><foreignObject><span data-marp-fitting-svg-content><span data-marp-fitting-svg-content-wrap><span class="hljs-comment"># Replaces Part 2.</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">complex_softmax</span>(<span class="hljs-params">x, eps=<span class="hljs-number">1e-7</span></span>):</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">reciprocal</span>(<span class="hljs-params">x</span>):</span>
<span class="hljs-keyword">return</span> x.conj() / (x * x.conj() + eps)
x2 = x - x[np.argmax(x.real)]
e = np.exp(x2)
<span class="hljs-keyword">return</span> e * reciprocal(np.<span class="hljs-built_in">sum</span>(e))
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">dss_kernel</span>(<span class="hljs-params">W, Lambda, L, step</span>):</span>
P = (step * Lambda)[:, <span class="hljs-literal">None</span>] * np.arange(L)
S = jax.vmap(complex_softmax)(P)
<span class="hljs-keyword">return</span> ((W / Lambda) @ S).ravel().real
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">dss_ssm</span>(<span class="hljs-params">W, Lambda, L, step</span>):</span>
N = Lambda.shape[<span class="hljs-number">0</span>]
Abar = np.diag(np.exp(Lambda * step))
b = jax.vmap(<span class="hljs-keyword">lambda</span> l:
<span class="hljs-number">1</span> / (l * (np.exp(l * np.arange(L) * step)).<span class="hljs-built_in">sum</span>()))
Bbar = b(Lambda).reshape(N, <span class="hljs-number">1</span>)
Cbar = W.reshape(<span class="hljs-number">1</span>, N)
<span class="hljs-keyword">return</span> (Abar, Bbar, Cbar)
</span></span></foreignObject></svg></code></pre>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="72" data-paginate="true" data-theme="default" data-marpit-pagination="72" data-marpit-pagination-total="72" style="--paginate:true;--theme:default;">
<h2>Thank You</h2>
<ul>
<li>
<p>Huge thanks to <a href="http://web.stanford.edu/~albertgu/">Albert Gu</a> and <a href="https://krandiash.github.io/">Karan Goel</a>, who were super helpful in putting this together. Their <a href="https://arxiv.org/abs/2111.00396">paper</a> and <a href="https://github.com/HazyResearch/state-spaces">codebase</a>.</p>
</li>
<li>
<p>Ankit Gupta for helping with his DSS model</p>
</li>
<li>
<p>Thanks to Conner Vercellino, Laurel Orr, Ankit Gupta, Ekin Akyürek, Saurav Maheshkar</p>
</li>
</ul>
</section>
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