Tensor-Tensor Product Toolbox

Introduction

The tensor-tensor product (t-product) [1] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD (see an illustration in the figure below), tensor spectral norm, tensor nuclear norm [2] and many others. The linear algebraic structure of tensors are similar to the matrix cases. We develop a Matlab toolbox to implement several basic operations on tensors based on t-product.

List of Functions

The table below gives the list of functions implemented in our toolbox. The detailed definitions of these tensor concepts, operations and tensor factorizations are given at https://canyilu.github.io/publications/2018-software-tproduct.pdf.

Note that we only focus on 3 way tensor in this toolbox. We will develop the same functions for p-way tensor in the near future. We will also provide the python version soon.

Simply run the following routine to test all the above functions:

test.m

Citing

In citing this toolbox in your papers, please use the following references:

C. Lu. Tensor-Tensor Product Toolbox. Carnegie Mellon University, June 2018. https://github.com/canyilu/tproduct.
C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin, and S. Yan. Tensor robust principal component analysis with a new tensor
nuclear norm. arXiv preprint arXiv:1804.03728, 2018.

The corresponding BiBTeX citations are given below:

@manual{lu2018tproduct,
  author       = {Lu, Canyi},
  title        = {Tensor-Tensor Product Toolbox},
  organization = {Carnegie Mellon University},
  month        = {June},
  year         = {2018},
  note         = {\url{https://github.com/canyilu/tproduct}}
}
@article{lu2018tensor,
  author       = {Lu, Canyi and Feng, Jiashi and Chen, Yudong and Liu, Wei and Lin, Zhouchen and Yan, Shuicheng},
  title        = {Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm},
  journal      = {arXiv preprint arXiv:1804.03728},
  year         = {2018}
}

Related Toolboxes

The t-product toolbox has been applied in our works about tensor roubst PCA [2,3], low-rank tensor completion and low-rank tensor recovery from Gaussian measurements [4]. Some more models are included in LibADMM toolbox [5].

• Tensor robust principal component analysis
• Low tubal tensor completion and tensor recovery from Gaussian measurements
• A Library of ADMM for Sparse and Low-rank Optimization

References

[1]	M. E. Kilmer and C. D. Martin. Factorization strategies for third-order tensors. Linear Algebra and its Applications. 435(3):641–658, 2011.
[2]	C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin, and S. Yan. Tensor robust principal component analysis with a new tensor nuclear norm. arXiv preprint arXiv:1804.03728, 2018.
[3]	C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin, and S. Yan. Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization. In IEEE International Conference on Computer Vision and Pattern Recognition, 2016.
[4]	C. Lu, J. Feng, Z. Lin, and S. Yan. Exact low tubal rank tensor recovery from Gaussian measurements. In International Joint Conference on Artificial Intelligence, 2018.
[5]	C. Lu, J. Feng, S. Yan, Z. Lin. A Unified Alternating Direction Method of Multipliers by Majorization Minimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 40, pp. 527-541, 2018.