README.md

May 1, 2021 · View on GitHub

Tensor-Tensor Product Toolbox 2.0 (updated in April, 2021)

1. T-product Toolbox 1.0

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 have tensor-tensor product, tensor SVD, tensor inverse and some other reated concepts extended from matrices. The detailed definitions of these tensor concepts, operations and tensor factorizations are given at https://canyilu.github.io/publications/2018-software-tproduct.pdf. We develop a Matlab toolbox to implement several basic operations on tensors based on t-product. See a list of implemented functions in t-product toolbox 1.0 below.

2. T-product Toolbox 2.0

The original t-product [1] uses the discrete Fourier transform and uses the fast Fourier transform (FFT) for efficient computing. It is further generlaized to the t-product under arbitrary invertible linear transform in [2]. Thus, all the concepts (e.g., tsvd, tensor inverse) of t-product under FFT can be generalized to t-product under general linear transforms. If the linear transform satisfies $L^\top L = LL^\top=\ell I$ where $\ell>0$ , then we can define a more general tensor nuclear norm induced by the t-product under this linear transform. Then we develop a more general Matlab toolbox to implement t-product under general linear transform. See a list of implemented functions in t-product toolbox 2.0 below.

For the definitions of t-product and related concepts under linear transform, please refer to [2] and our works [6,7]. We will provide a document to give the details in the future.

3. Examples

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

test.m

4. Citation

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. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2019.
C. Lu, X. Peng, and Y. Wei. Low-Rank Tensor Completion With a New Tensor Nuclear Norm Induced by Invertible Linear Transforms. IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2019

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      = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
  year         = {2019}
}
@inproceedings{lu2019tensor,
  author       = {Lu, Canyi and Peng, Xi and Wei, Yunchao},
  title        = {Low-Rank Tensor Completion With a New Tensor Nuclear Norm Induced by Invertible Linear Transforms},
  journal      = {CVPR},
  year         = {2019}
}

5. Version History

Version 1.0 was released on June, 2018. It implements the functions of t-product and related concepts under fast Fourier transform.
Version 2.0 was released on April, 2021. It implements the functions of t-product and related concepts under general invertible linear transform. The fast Fourier transform is the default transform.
- Most functions are direct generalization from the fast Fourier transform to general linear transform, e.g., tprod, tran, teye, tinv, tsvd, tubalrank, tsn, tnn, prox_tnn and tqr.
- Some functions are new (not included in Version 1.0), e.g., basis_column, basis_tube and unit_eijk.
- Some functions in Version 1.0 are updated, e.g., the setting of parameter tol in tubalrank and tsvd is updated, and tprod, tsn, tinv and tqr are updated.

The t-product toolbox has been applied in our works for tensor roubst PCA [3,4], low-rank tensor completion and low-rank tensor recovery from Gaussian measurements [5]. The t-product under linear transform has also been applied in tensor completion [6] and tensor robust PCA [7]. Some more models are included in LibADMM toolbox [8].

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]	M. E. Kilmer and S. Aeron. Tensor-Tensor Products with Invertible Linear Transforms. Linear Algebra and its Applications. 2015.
[3]	C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin, and S. Yan. Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2019.
[4]	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.
[5]	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.
[6]	C. Lu, X. Peng, and Y. Wei. Low-Rank Tensor Completion With a New Tensor Nuclear Norm Induced by Invertible Linear Transforms. IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
[7]	C. Lu. Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms. arXiv preprint arXiv:1907.08288. 2019.
[8]	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.