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 where , 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.

6. Related Toolboxes

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].

• Tensor robust principal component analysis
• Low tubal tensor completion and tensor recovery from Gaussian measurements
• Tensor robust PCA and tensor completion based on tensor nuclear norm under linear transform
• 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]	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.