Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms

This work studies the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is motivated by the recently proposed linear transforms based tensor-tensor product and tensor SVD. We define a new transforms depended tensor rank and the corresponding tensor nuclear norm. Then we solve the TRPCA problem by convex optimization whose objective is a weighted combination of the new tensor nuclear norm and the $\ell_1$-norm. In theory, we show that under certain incoherence conditions, the convex program exactly recovers the underlying low-rank and sparse components. It is of great interest that our new TRPCA model generalizes existing works. In particular, if the studied tensor reduces to a matrix, our TRPCA model reduces to the known matrix RPCA. Our new TRPCA which is allowed to use general linear transforms can be regarded as an extension of our former TRPCA work which uses the discrete Fourier transform. But their proof of the recovery guarantee is different. Numerical experiments verify our results and the application on image recovery demonstrates the superiority of our method.Comment: arXiv admin note: text overlap with arXiv:1804.03728; text overlap with arXiv:1311.6182 by other author

Robust Low-Rank Tensor Ring Completion

Low-rank tensor completion recovers missing entries based on different tensor decompositions. Due to its outstanding performance in exploiting some higher-order data structure, low rank tensor ring has been applied in tensor completion. To further deal with its sensitivity to sparse component as it does in tensor principle component analysis, we propose robust tensor ring completion (RTRC), which separates latent low-rank tensor component from sparse component with limited number of measurements. The low rank tensor component is constrained by the weighted sum of nuclear norms of its balanced unfoldings, while the sparse component is regularized by its l1 norm. We analyze the RTRC model and gives the exact recovery guarantee. The alternating direction method of multipliers is used to divide the problem into several sub-problems with fast solutions. In numerical experiments, we verify the recovery condition of the proposed method on synthetic data, and show the proposed method outperforms the state-of-the-art ones in terms of both accuracy and computational complexity in a number of real-world data based tasks, i.e., light-field image recovery, shadow removal in face images, and background extraction in color video

Color Image and Multispectral Image Denoising Using Block Diagonal Representation

Filtering images of more than one channel is challenging in terms of both efficiency and effectiveness. By grouping similar patches to utilize the self-similarity and sparse linear approximation of natural images, recent nonlocal and transform-domain methods have been widely used in color and multispectral image (MSI) denoising. Many related methods focus on the modeling of group level correlation to enhance sparsity, which often resorts to a recursive strategy with a large number of similar patches. The importance of the patch level representation is understated. In this paper, we mainly investigate the influence and potential of representation at patch level by considering a general formulation with block diagonal matrix. We further show that by training a proper global patch basis, along with a local principal component analysis transform in the grouping dimension, a simple transform-threshold-inverse method could produce very competitive results. Fast implementation is also developed to reduce computational complexity. Extensive experiments on both simulated and real datasets demonstrate its robustness, effectiveness and efficiency

Non-convex Penalty for Tensor Completion and Robust PCA

In this paper, we propose a novel non-convex tensor rank surrogate function and a novel non-convex sparsity measure for tensor. The basic idea is to sidestep the bias of $\ell_1-$norm by introducing concavity. Furthermore, we employ the proposed non-convex penalties in tensor recovery problems such as tensor completion and tensor robust principal component analysis, which has various real applications such as image inpainting and denoising. Due to the concavity, the models are difficult to solve. To tackle this problem, we devise majorization minimization algorithms, which optimize upper bounds of original functions in each iteration, and every sub-problem is solved by alternating direction multiplier method. Finally, experimental results on natural images and hyperspectral images demonstrate the effectiveness and efficiency of the proposed methods

Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.Comment: arXiv admin note: text overlap with arXiv:1708.0418

Robust Tensor Completion Using Transformed Tensor SVD

In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in PSNR than that by using Fourier transform and other robust tensor completion methods

Exploiting the structure effectively and efficiently in low rank matrix recovery

Low rank model arises from a wide range of applications, including machine learning, signal processing, computer algebra, computer vision, and imaging science. Low rank matrix recovery is about reconstructing a low rank matrix from incomplete measurements. In this survey we review recent developments on low rank matrix recovery, focusing on three typical scenarios: matrix sensing, matrix completion and phase retrieval. An overview of effective and efficient approaches for the problem is given, including nuclear norm minimization, projected gradient descent based on matrix factorization, and Riemannian optimization based on the embedded manifold of low rank matrices. Numerical recipes of different approaches are emphasized while accompanied by the corresponding theoretical recovery guarantees

Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion

The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods

Fast Randomized Singular Value Thresholding for Low-rank Optimization

Rank minimization can be converted into tractable surrogate problems, such as Nuclear Norm Minimization (NNM) and Weighted NNM (WNNM). The problems related to NNM, or WNNM, can be solved iteratively by applying a closed-form proximal operator, called Singular Value Thresholding (SVT), or Weighted SVT, but they suffer from high computational cost of Singular Value Decomposition (SVD) at each iteration. We propose a fast and accurate approximation method for SVT, that we call fast randomized SVT (FRSVT), with which we avoid direct computation of SVD. The key idea is to extract an approximate basis for the range of the matrix from its compressed matrix. Given the basis, we compute partial singular values of the original matrix from the small factored matrix. In addition, by developping a range propagation method, our method further speeds up the extraction of approximate basis at each iteration. Our theoretical analysis shows the relationship between the approximation bound of SVD and its effect to NNM via SVT. Along with the analysis, our empirical results quantitatively and qualitatively show that our approximation rarely harms the convergence of the host algorithms. We assess the efficiency and accuracy of the proposed method on various computer vision problems, e.g., subspace clustering, weather artifact removal, and simultaneous multi-image alignment and rectification.Comment: Appeared in CVPR 2015, and under major revision of TPAMI. Source code is available on http://thoh.kaist.ac.k

Frequency-Weighted Robust Tensor Principal Component Analysis

Robust tensor principal component analysis (RTPCA) can separate the low-rank component and sparse component from multidimensional data, which has been used successfully in several image applications. Its performance varies with different kinds of tensor decompositions, and the tensor singular value decomposition (t-SVD) is a popularly selected one. The standard t-SVD takes the discrete Fourier transform to exploit the residual in the 3rd mode in the decomposition. When minimizing the tensor nuclear norm related to t-SVD, all the frontal slices in frequency domain are optimized equally. In this paper, we incorporate frequency component analysis into t-SVD to enhance the RTPCA performance. Specially, different frequency bands are unequally weighted with respect to the corresponding physical meanings, and the frequency-weighted tensor nuclear norm can be obtained. Accordingly we rigorously deduce the frequency-weighted tensor singular value threshold operator, and apply it for low rank approximation subproblem in RTPCA. The newly obtained frequency-weighted RTPCA can be solved by alternating direction method of multipliers, and it is the first time that frequency analysis is taken in tensor principal component analysis. Numerical experiments on synthetic 3D data, color image denoising and background modeling verify that the proposed method outperforms the state-of-the-art algorithms both in accuracy and computational complexity