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Add tfp.experimental.psd_kernels.AdditiveKernel.
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tensorflow_probability/python/experimental/psd_kernels/additive_kernel.py
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# Copyright 2021 The TensorFlow Probability Authors. | ||
# | ||
# Licensed under the Apache License, Version 2.0 (the "License"); | ||
# you may not use this file except in compliance with the License. | ||
# You may obtain a copy of the License at | ||
# | ||
# http://www.apache.org/licenses/LICENSE-2.0 | ||
# | ||
# Unless required by applicable law or agreed to in writing, software | ||
# distributed under the License is distributed on an "AS IS" BASIS, | ||
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. | ||
# See the License for the specific language governing permissions and | ||
# limitations under the License. | ||
# ============================================================================ | ||
"""Additive kernel.""" | ||
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import tensorflow.compat.v2 as tf | ||
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from tensorflow_probability.python.internal import dtype_util | ||
from tensorflow_probability.python.internal import parameter_properties | ||
from tensorflow_probability.python.internal import prefer_static as ps | ||
from tensorflow_probability.python.internal import tensor_util | ||
from tensorflow_probability.python.math.psd_kernels import positive_semidefinite_kernel as psd_kernel | ||
from tensorflow_probability.python.math.psd_kernels.internal import util | ||
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__all__ = [ | ||
'AdditiveKernel', | ||
] | ||
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class AdditiveKernel(psd_kernel.AutoCompositeTensorPsdKernel): | ||
"""Additive Kernel. | ||
This kernel has the following form | ||
```none | ||
k(x, y) = sum k_add_i(x, y) | ||
k_add_n(x, y) = a_n**2 sum_{1<=i1<i2<...in} prod k_i(x[i], y[i]) | ||
``` | ||
Where $k_i$ is the one-dimensional base kernel for the `i`th dimension. | ||
In other words, this computes sums of elementary symmetric polynomials | ||
over `k_i(x[i], y[i])`. | ||
This kernel is very related to the ANOVA kernel defined as: | ||
`k_{ANOVA}(x, y) = prod (1 + k_i(x[i], x[i])`. `k_{ANOVA}` is | ||
equivalent to a special case of this kernel where the `amplitudes` are | ||
all one, along with a constant shift by 1. | ||
#### References | ||
[1] D. Duvenaud, H. Nickish, C. E. Rasmussen, Additive Gaussian Process. | ||
https://hannes.nickisch.org/papers/conferences/duvenaud11gpadditive.pdf | ||
[2] M. Stitson, A. Gammerman, V. Vapnik, V. Vovk, et al. | ||
Support Vector Regression with ANOVA Decomposition Kernels | ||
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.7818&rep=rep1&type=pdf | ||
""" | ||
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def __init__( | ||
self, | ||
kernel, | ||
amplitudes, | ||
validate_args=False, | ||
name='AdditiveKernel'): | ||
"""Instantiates an `AdditiveKernel`. | ||
Args: | ||
kernel: An instance of `PositiveSemidefiniteKernel`s that are defined | ||
within this class (specifically they allow for reinterpreting | ||
batch dimensions as feature dimensions) that act on inputs of | ||
the form `[B1, ...., Bk, D, 1]`; that is, `kernel` is a batch of | ||
D-kernels, each acting on 1-dimensional inputs. We assume that the | ||
kernel has a batch dimension broadcastable with `[D]`. `kernel` must | ||
inherit from `tf.__internal__.CompositeTensor`. | ||
amplitudes: `Tensor` of shape `[B1, ...., Bk, M]`, where `M` is the order | ||
of the additive kernel. `M` must be statically identifiable. | ||
validate_args: Python `bool`, default `False`. When `True` kernel | ||
parameters are checked for validity despite possibly degrading runtime | ||
performance. When `False` invalid inputs may silently render incorrect | ||
outputs. | ||
name: Python `str` name prefixed to Ops created by this class. Default: | ||
subclass name. | ||
Raises: | ||
TypeError: if `kernel` is not an instance of | ||
`tf.__internal__.CompositeTensor`. | ||
""" | ||
parameters = dict(locals()) | ||
with tf.name_scope(name): | ||
if not isinstance(kernel, tf.__internal__.CompositeTensor): | ||
raise TypeError('`kernel` must inherit from ' | ||
'`tf.__internal__.CompositeTensor`.') | ||
dtype = util.maybe_get_common_dtype([kernel, amplitudes]) | ||
self._kernel = kernel | ||
self._amplitudes = tensor_util.convert_nonref_to_tensor( | ||
amplitudes, name='amplitudes', dtype=dtype) | ||
super(AdditiveKernel, self).__init__( | ||
feature_ndims=self.kernel.feature_ndims, | ||
dtype=dtype, | ||
name=name, | ||
validate_args=validate_args, | ||
parameters=parameters) | ||
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@property | ||
def amplitudes(self): | ||
"""Amplitude parameter for each additive kernel.""" | ||
return self._amplitudes | ||
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@property | ||
def kernel(self): | ||
"""Inner kernel used for scalar kernel computations.""" | ||
return self._kernel | ||
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@classmethod | ||
def _parameter_properties(cls, dtype, num_classes=None): | ||
from tensorflow_probability.python.bijectors import softplus as softplus_bijector # pylint:disable=g-import-not-at-top | ||
return dict( | ||
amplitudes=parameter_properties.ParameterProperties( | ||
event_ndims=1, | ||
default_constraining_bijector_fn=( | ||
softplus_bijector.Softplus(low=dtype_util.eps(dtype)))), | ||
kernel=parameter_properties.BatchedComponentProperties(event_ndims=1)) | ||
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# Below, the Additive Kernel is computed via a recurrence on elementary | ||
# symmetric polynomials. | ||
# Let z_i = k[i](x[i], y[i]) | ||
# Then we are computing the elementary symmetric polynomials | ||
# S_n(z_1, ..., z_k) = \sum_i \prod_{1 <= j_1 < j_2, ... < j_n <= k} z_j | ||
# Elementary symmetric polynomials satisfy the recurrence: | ||
# S_n(z_1, ..., z_k) = S_n(z_1, ..., z_{k-1}) + | ||
# z_k * S_{n - 1}(z_1, ..., z_{k - 1}) | ||
# Thus, we can use dynamic programming to compute the elementary symmetric | ||
# polynomials over z_i, and use vectorization to do this in a batched way. | ||
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def _apply(self, x1, x2, example_ndims=0): | ||
@tf.recompute_grad | ||
def _inner_apply(x1, x2): | ||
order = ps.shape(self.amplitudes)[-1] | ||
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def scan_fn(esp, i): | ||
s = self.kernel[..., i].apply( | ||
x1[..., i][..., tf.newaxis], | ||
x2[..., i][..., tf.newaxis], | ||
example_ndims=example_ndims) | ||
next_esp = esp[..., 1:] + s[..., tf.newaxis] * esp[..., :-1] | ||
# Add the zero-th polynomial. | ||
next_esp = tf.concat( | ||
[tf.ones_like(esp[..., 0][..., tf.newaxis]), next_esp], axis=-1) | ||
return next_esp | ||
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batch_shape = ps.broadcast_shape( | ||
ps.shape(x1)[:-self.kernel.feature_ndims], | ||
ps.shape(x2)[:-self.kernel.feature_ndims]) | ||
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batch_shape = ps.broadcast_shape( | ||
batch_shape, | ||
ps.concat([ | ||
self.batch_shape_tensor(), | ||
[1] * example_ndims], axis=0)) | ||
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initializer = tf.concat( | ||
[tf.ones(ps.concat([batch_shape, [1]], axis=0), | ||
dtype=self.dtype), | ||
tf.zeros(ps.concat([batch_shape, [order]], axis=0), | ||
dtype=self.dtype)], axis=-1) | ||
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esps = tf.scan( | ||
scan_fn, | ||
elems=ps.range(0, ps.shape(x1)[-1], dtype=tf.int32), | ||
parallel_iterations=32, | ||
initializer=initializer)[-1, ..., 1:] | ||
amplitudes = util.pad_shape_with_ones( | ||
self.amplitudes, ndims=example_ndims, start=-2) | ||
return tf.reduce_sum(esps * tf.math.square(amplitudes), axis=-1) | ||
return _inner_apply(x1, x2) | ||
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def _matrix(self, x1, x2): | ||
@tf.recompute_grad | ||
def _inner_matrix(x1, x2): | ||
order = ps.shape(self.amplitudes)[-1] | ||
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def scan_fn(esp, i): | ||
s = self.kernel[..., i].matrix( | ||
x1[..., i][..., tf.newaxis], x2[..., i][..., tf.newaxis]) | ||
next_esp = esp[..., 1:] + s[..., tf.newaxis] * esp[..., :-1] | ||
# Add the zero-th polynomial. | ||
next_esp = tf.concat( | ||
[tf.ones_like(esp[..., 0][..., tf.newaxis]), next_esp], axis=-1) | ||
return next_esp | ||
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batch_shape = ps.broadcast_shape( | ||
ps.shape(x1)[:-(self.kernel.feature_ndims + 1)], | ||
ps.shape(x2)[:-(self.kernel.feature_ndims + 1)]) | ||
batch_shape = ps.broadcast_shape( | ||
batch_shape, self.batch_shape_tensor()) | ||
matrix_shape = [ | ||
ps.shape(x1)[-(self.kernel.feature_ndims + 1)], | ||
ps.shape(x2)[-(self.kernel.feature_ndims + 1)]] | ||
total_shape = ps.concat([batch_shape, matrix_shape], axis=0) | ||
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initializer = tf.concat( | ||
[tf.ones(ps.concat([total_shape, [1]], axis=0), | ||
dtype=self.dtype), | ||
tf.zeros(ps.concat([total_shape, [order]], axis=0), | ||
dtype=self.dtype)], axis=-1) | ||
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esps = tf.scan( | ||
scan_fn, | ||
elems=ps.range(0, ps.shape(x1)[-1], dtype=tf.int32), | ||
parallel_iterations=32, | ||
initializer=initializer)[-1, ..., 1:] | ||
amplitudes = util.pad_shape_with_ones( | ||
self.amplitudes, ndims=2, start=-2) | ||
return tf.reduce_sum(esps * tf.math.square(amplitudes), axis=-1) | ||
return _inner_matrix(x1, x2) |
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