Parallel matrix factorization for low-rank tensor completion

By Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. arXiv.org, 2013

388 citations recorded by the source corpus.

Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.   Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.

Subfields: Computational Mathematics, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/538685e2feb6c59f90034583f615b150d92acb84

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Zhouchen Lin
2. Canyi Lu
3. Deyu Meng
4. Qi Xie
5. Qian Zhao
6. Zongben Xu
7. John Wright
8. Hoang Duong Tuan

In Computational Mechanics

1. Zhouchen Lin
2. Canyi Lu
3. Deyu Meng
4. Zongben Xu
5. Qi Xie
6. Qian Zhao
7. Minh N. Do
8. Donald Goldfarb

In Computer Vision and Pattern Recognition

1. Deyu Meng
2. Zongben Xu
3. Qian Zhao
4. Zhouchen Lin
5. Ming-Hsuan Yang
6. Qi Xie
7. Canyi Lu
8. Luc Van Gool

Citations in the corpus

• Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery (2013, methodology): “In (Romera-Paredes et al., 2013; Xu et al., 2013), efficient methods based on alternating minimization are designed to solve related non-convex models.”
• Tensor Factorization for Low-Rank Tensor Completion (2018, methodology): “Recent research [2], [3], [6], [7], [10]–[13] shows that high dimensional tensor data of interest, such as videos and image collections, are usually intrinsically low-rank or approximately so.”
• Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery (2018, methodology): “The comparison methods including: ADMM(ALM)-based matrix completion (MC-ALM) [36], HaLRTC [40], the factorization-based TC method (TMac) [70], joint trace/TV based TC method (Trace/TV) [23], ten-sor-SVD-based TC method (t-SVD) [74], minmax concave plus penalty-based TC method (McpTC) [11] and the…”
• Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train (2016, background): “Specifically, if the SVD of X [ k ] = UλV T , its thresholding SVD is defined as: where λ γ k = diag (max( λ l − γ k , 0)) .”