Nonconvex tensor rank minimization and its applications to tensor recovery

Abstract Low-rank tensor recovery (LRTR) has recently emerged as the potent tools for representing multidimensional data. One of the most popular LRTR is tensor rank minimization that acts as the estimating tensor rank for a given tensor. However, the existing convex tensor rank approximation methods suffer from the serious rank estimation bias due to neglecting the physical meanings of singular values along each mode. In this paper, we propose a new method to approximate the tensor rank by using the nonconvex logarithmic surrogate function of the singular values, and the redefined rank approximation can further reduce to a convex weighted nuclear norm minimization (WNNM) problem. By embedding the tensor rank function into the tensor completion (TC) and tensor robust PCA (TRPCA) frameworks, new models are formulated to enhance tensor processing. Additionally, by introducing relaxation forms of the proposed tensor rank function, the alternating direction method of multipliers (ADMM) can be adopted for the models. The proposed nonconvex tensor rank minimization method can achieve state-of-the-art performance in tensor recovery, including tensor completion and background subtraction.

[1]  Andrzej Cichocki,et al.  Total Variation Regularized Tensor RPCA for Background Subtraction From Compressive Measurements , 2015, IEEE Transactions on Image Processing.

[2]  Jonathan Cheung-Wai Chan,et al.  Nonlocal Low-Rank Regularized Tensor Decomposition for Hyperspectral Image Denoising , 2019, IEEE Transactions on Geoscience and Remote Sensing.

[3]  Yueting Zhuang,et al.  Tensor-Based Transductive Learning for Multimodality Video Semantic Concept Detection , 2009, IEEE Transactions on Multimedia.

[4]  Qi Xie,et al.  A Novel Sparsity Measure for Tensor Recovery , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[5]  Jian Yang,et al.  Ieee Transactions on Image Processing 1 Tensor Discriminant Color Space for Face Recognition , 2022 .

[6]  Yilong Yin,et al.  Distribution-Oriented Aesthetics Assessment With Semantic-Aware Hybrid Network , 2019, IEEE Transactions on Multimedia.

[7]  Zenglin Xu,et al.  Bayesian Nonparametric Models for Multiway Data Analysis , 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  Meng Liu,et al.  Online Data Organizer: Micro-Video Categorization by Structure-Guided Multimodal Dictionary Learning , 2019, IEEE Transactions on Image Processing.

[9]  Donald Goldfarb,et al.  Robust Low-Rank Tensor Recovery: Models and Algorithms , 2013, SIAM J. Matrix Anal. Appl..

[10]  Qi Xie,et al.  Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery , 2018, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[12]  M. Kilmer,et al.  Factorization strategies for third-order tensors , 2011 .

[13]  Lei Zhang,et al.  Weighted Nuclear Norm Minimization with Application to Image Denoising , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[14]  Zemin Zhang,et al.  Exact Tensor Completion Using t-SVD , 2015, IEEE Transactions on Signal Processing.

[15]  Erik Cambria,et al.  Tensor Fusion Network for Multimodal Sentiment Analysis , 2017, EMNLP.

[16]  Dong Liang,et al.  Tensor RPCA by Bayesian CP Factorization with Complex Noise , 2017, 2017 IEEE International Conference on Computer Vision (ICCV).

[17]  Hailong Sun,et al.  Temporal QoS-aware web service recommendation via non-negative tensor factorization , 2014, WWW.

[18]  Yandong Tang,et al.  A Generalized Model for Robust Tensor Factorization With Noise Modeling by Mixture of Gaussians , 2018, IEEE Transactions on Neural Networks and Learning Systems.

[19]  Dongdai Lin,et al.  Robust Face Clustering Via Tensor Decomposition , 2015, IEEE Transactions on Cybernetics.

[20]  Deyu Meng,et al.  Robust Matrix Factorization with Unknown Noise , 2013, 2013 IEEE International Conference on Computer Vision.

[21]  Andrzej Cichocki,et al.  Smooth PARAFAC Decomposition for Tensor Completion , 2015, IEEE Transactions on Signal Processing.

[22]  Stéphane Chrétien,et al.  Von Neumann's trace inequality for tensors , 2015 .

[23]  Tamara G. Kolda,et al.  Scalable Tensor Factorizations for Incomplete Data , 2010, ArXiv.

[24]  Jieping Ye,et al.  Tensor Completion for Estimating Missing Values in Visual Data , 2013, IEEE Trans. Pattern Anal. Mach. Intell..

[25]  Hong-Yuan Mark Liao,et al.  Simultaneous Tensor Decomposition and Completion Using Factor Priors , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[26]  Lei Zhang,et al.  Robust Principal Component Analysis with Complex Noise , 2014, ICML.

[27]  Liqing Zhang,et al.  Bayesian Robust Tensor Factorization for Incomplete Multiway Data , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[28]  Ming Yuan,et al.  On Tensor Completion via Nuclear Norm Minimization , 2014, Foundations of Computational Mathematics.

[29]  Wei Liu,et al.  Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[30]  Marko Filipovic,et al.  Tucker factorization with missing data with application to low-$$n$$n-rank tensor completion , 2015, Multidimens. Syst. Signal Process..

[31]  Eric L. Miller,et al.  Tensor-Based Formulation and Nuclear Norm Regularization for Multienergy Computed Tomography , 2013, IEEE Transactions on Image Processing.

[32]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[33]  M. Wegkamp,et al.  Joint variable and rank selection for parsimonious estimation of high-dimensional matrices , 2011, 1110.3556.

[34]  Meng Liu,et al.  Attentive Moment Retrieval in Videos , 2018, SIGIR.

[35]  Dapeng Tao,et al.  Tensor Manifold Discriminant Projections for Acceleration-Based Human Activity Recognition , 2016, IEEE Transactions on Multimedia.

[36]  Shuicheng Yan,et al.  Practical low-rank matrix approximation under robust L1-norm , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[37]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[38]  Bo Huang,et al.  Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery , 2013, ICML.

[39]  Andrzej Cichocki,et al.  Robust Multilinear Tensor Rank Estimation Using Higher Order Singular Value Decomposition and Information Criteria , 2017, IEEE Transactions on Signal Processing.

[40]  Eero P. Simoncelli,et al.  Image quality assessment: from error visibility to structural similarity , 2004, IEEE Transactions on Image Processing.

[41]  Liqing Zhang,et al.  Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[42]  Ming-Hsuan Yang,et al.  Generative Face Completion , 2017, 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[43]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[44]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[45]  Jonathan Cheung-Wai Chan,et al.  Nonlocal Tensor Sparse Representation and Low-Rank Regularization for Hyperspectral Image Compressive Sensing Reconstruction , 2019, Remote. Sens..

[46]  Jiayu Zhou,et al.  Who, What, When, and Where: Multi-Dimensional Collaborative Recommendations Using Tensor Factorization on Sparse User-Generated Data , 2015, WWW.

[47]  Wotao Yin,et al.  Parallel matrix factorization for low-rank tensor completion , 2013, ArXiv.

[48]  Ryota Tomioka,et al.  Estimation of low-rank tensors via convex optimization , 2010, 1010.0789.

[49]  B. Recht,et al.  Tensor completion and low-n-rank tensor recovery via convex optimization , 2011 .