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Exact Tensor Completion Powered by Slim Transforms
Published 2 Feb 2024 in cs.LG and eess.SP | (2402.03468v2)
Abstract: In this work, a tensor completion problem is studied, which aims to perfectly recover the tensor from partial observations. The existing theoretical guarantee requires the involved transform to be orthogonal, which hinders its applications. In this paper, jumping out of the constraints of isotropy and self-adjointness, the theoretical guarantee of exact tensor completion with arbitrary linear transforms is established by directly operating the tensors in the transform domain. With the enriched choices of transforms, a new analysis obtained by the proof discloses why slim transforms outperform their square counterparts from a theoretical level. Our model and proof greatly enhance the flexibility of tensor completion and extensive experiments validate the superiority of the proposed method.
- Y. Panagakis, J. Kossaifi, G. G. Chrysos, J. Oldfield, M. A. Nicolaou, A. Anandkumar, and S. Zafeiriou, “Tensor methods in computer vision and deep learning,” Proceedings of the IEEE, vol. 109, no. 5, pp. 863–890, 2021.
- N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E. Papalexakis, and C. Faloutsos, “Tensor decomposition for signal processing and machine learning,” IEEE Transactions on Signal Processing, vol. 65, no. 13, pp. 3551–3582, 2017.
- S. Rabanser, O. Shchur, and S. Günnemann, “Introduction to tensor decompositions and their applications in machine learning,” 2017.
- H. Yin, “Tensor sparse representation for 3-d medical image fusion using weighted average rule,” IEEE Transactions on Biomedical Engineering, vol. 65, no. 11, pp. 2622–2633, 2018.
- Y. Li, W. Liang, K. Xie, D. Zhang, S. Xie, and K. Li, “Lightnestle: Quick and accurate neural sequential tensor completion via meta learning,” in IEEE INFOCOM 2023 - IEEE Conference on Computer Communications, pp. 1–10, 2023.
- T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM Review, vol. 51, no. 3, pp. 455–500, 2009.
- L. De Lathauwer and B. De Moor, “From matrix to tensor: Multilinear algebra and signal processing,” in Institute of mathematics and its applications conference series, vol. 67, pp. 1–16, Citeseer, 1998.
- Y. Chen, A. Jalali, S. Sanghavi, and C. Caramanis, “Low-rank matrix recovery from errors and erasures,” IEEE Transactions on Information Theory, vol. 59, no. 7, pp. 4324–4337, 2013.
- E. J. Candès and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Transactions on Information Theory, vol. 56, no. 5, pp. 2053–2080, 2010.
- E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of the ACM (JACM), vol. 58, no. 3, pp. 1–37, 2011.
- J. Liu, P. Musialski, P. Wonka, and J. Ye, “Tensor completion for estimating missing values in visual data,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 1, pp. 208–220, 2013.
- I. V. Oseledets, “Tensor-train decomposition,” SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2295–2317, 2011.
- Q. Zhao, G. Zhou, S. Xie, L. Zhang, and A. Cichocki, “Tensor ring decomposition,” arXiv preprint arXiv:1606.05535, 2016.
- Z. Zhang and S. Aeron, “Exact tensor completion using t-svd,” IEEE Transactions on Signal Processing, vol. 65, no. 6, pp. 1511–1526, 2017.
- W. Qin, H. Wang, F. Zhang, J. Wang, X. Luo, and T. Huang, “Low-rank high-order tensor completion with applications in visual data,” IEEE Transactions on Image Processing, vol. 31, pp. 2433–2448, 2022.
- 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, vol. 42, no. 4, pp. 925–938, 2020.
- C. Lu, X. Peng, and Y. Wei, “Low-rank tensor completion with a new tensor nuclear norm induced by invertible linear transforms,” in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5989–5997, 2019.
- C. Lu, “Transforms based tensor robust pca: Corrupted low-rank tensors recovery via convex optimization,” in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 1125–1132, 2021.
- T.-X. Jiang, M. K. Ng, X.-L. Zhao, and T.-Z. Huang, “Framelet representation of tensor nuclear norm for third-order tensor completion,” IEEE Transactions on Image Processing, vol. 29, pp. 7233–7244, 2020.
- Y.-S. Luo, X.-L. Zhao, T.-X. Jiang, Y. Chang, M. K. Ng, and C. Li, “Self-supervised nonlinear transform-based tensor nuclear norm for multi-dimensional image recovery,” IEEE Transactions on Image Processing, vol. 31, pp. 3793–3808, 2022.
- T.-X. Jiang, X.-L. Zhao, H. Zhang, and M. K. Ng, “Dictionary learning with low-rank coding coefficients for tensor completion,” IEEE Transactions on Neural Networks and Learning Systems, vol. 34, no. 2, pp. 932–946, 2023.
- K. Lee, Y. Li, K. H. Jin, and J. C. Ye, “Unified theory for recovery of sparse signals in a general transform domain,” IEEE Transactions on Information Theory, vol. 64, no. 8, pp. 5457–5477, 2018.
- J. A. Tropp, “User-friendly tail bounds for sums of random matrices,” Foundations of computational mathematics, vol. 12, pp. 389–434, 2012.
- P. Korohoda and A. Dąbrowski, “Generalized convolution concept based on dct,” in 2004 12th European Signal Processing Conference, pp. 973–976, 2004.
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