No abstract

Low-rank tensor completion (LRTC) has successfully been applied to a wide range of real-world problems. Despite the broad, successful applications, existing LRTC methods may become very slow or even not applicable for large-scale problems. To address this issue, a novel core tensor trace-norm minimization (CTNM) method is proposed for simultaneous tensor learning and decomposition, and has a much lower computational complexity. In our solution, first, the equivalence relation of trace norm of a low-rank tensor and its core tensor is induced. Second, the trace norm of the core tensor is used to replace that of the whole tensor, which leads to two much smaller scale matrix TNM problems. Finally, an efficient alternating direction augmented Lagrangian method is developed to solve our problems. Our CTNM formulation needs only O((R+NRI)log(√{I})) observations to reliably recover an N th-order I×I×…×I tensor of n -rank (r,r,…,r) , compared with O(rI) observations required by those tensor TNM methods ( I >> R ≥ r ). Extensive experimental results show that CTNM is usually more accurate than them, and is orders of magnitude faster.

The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to $\ell _{p}$ -norm minimization with two specific values of $p$ , i.e., $p=1/2$ and $p=2/3$ , we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten- $1/2$ and $2/3$ quasi-norms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our methods yield more accurate solutions than original Schatten quasi-norm minimization, even when the number of observations is very limited. Finally, we apply our penalties to various low-level vision problems, e.g., text removal, moving object detection, image alignment and inpainting, and show that our methods usually outperform the state-of-the-art methods.

In recent years, low-rank tensor completion (LRTC) problems have received a significant amount of attention in computer vision, data mining, and signal processing. The existing trace norm minimization algorithms for iteratively solving LRTC problems involve multiple singular value decompositions of very large matrices at each iteration. Therefore, they suffer from high computational cost. In this paper, we propose a novel trace norm regularized CANDECOMP/PARAFAC decomposition (TNCP) method for simultaneous tensor decomposition and completion. We first formulate a factor matrix rank minimization model by deducing the relation between the rank of each factor matrix and the mode- n rank of a tensor. Then, we introduce a tractable relaxation of our rank function, and then achieve a convex combination problem of much smaller-scale matrix trace norm minimization. Finally, we develop an efficient algorithm based on alternating direction method of multipliers to solve our problem. The promising experimental results on synthetic and real-world data validate the effectiveness of our TNCP method. Moreover, TNCP is significantly faster than the state-of-the-art methods and scales to larger problems.

Recovering matrices from incomplete and corrupted observations is a fundamental problem with many applications in various areas of science and engineering. In theory, under certain conditions, this problem can be solved via a natural convex relaxation. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large scale problems. In this paper, we propose a robust principal component analysis (RPCA) plus matrix completion framework to recover low-rank and sparse matrices from missing and grossly corrupted observations. Under the unified framework, we first present a convex robust matrix completion model to replace the linear projection operator constraint by a simple equality one. To further improve the efficiency of our convex model, we also develop a scalable structured factorization model, which can yield an orthogonal dictionary and a robust data representation simultaneously. Then, we develop two alternating direction augmented Lagrangian (ADAL) algorithms to efficiently solve the proposed problems. Finally, we discuss the convergence analysis of our algorithms. Experimental results verified both the efficiency and effectiveness of our methods compared with the state-of-the-art algorithms.