张量积工具箱(t-product toolkit)-Matlab实现

介绍

张量积(t-product)[1]是矩阵乘法的一种自然泛化。基于t-product,矩阵上的许多操作可以扩展到张量情况,包括张量SVD(参见下图中的图例),张量谱范数,张量核范数[2]等等。张量的线性代数结构与矩阵情况类似。我们开发一个Matlab工具箱来实现几个基于t-product的张量基本操作。

功能列表

下表列出了我们工具箱中实现的功能列表。这些张量概念,操作和张量分解的详细定义在下面给出https://canyilu.github.io/publications/2018-software-tproduct.pdf

请注意,我们只关注这个工具箱中的三维张量。我们将在不久的将来为p路张量开发相同的功能。我们也将很快提供python版本。

只需运行以下例程即可测试以上所有功能:

test.m

Related Toolboxes相关工具箱

t-product 工具箱已经应用于我们的工作张量鲁棒PCA(tensor roubst PCA) [2,3], low-rank tensor completion and 高斯测量低秩张量恢复(low-rank tensor recovery from Gaussian measurements) [4]. 更多的模型在 LibADMM toolbox [5].

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.
https://github.com/canyilu/tproduct

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