Recovering Low-Rank and Sparse Matrices via Robust Bilateral Factorization

Abstract

Recovering low-rank and sparse matrices from partial, incomplete or corrupted observations is an important problem in many areas of science and engineering. In this paper, we propose a scalable robust bilateral factorization (RBF) method to recover both structured matrices from missing and grossly corrupted data such as robust matrix completion (RMC), or incomplete and grossly corrupted measurements such as compressive principal component pursuit (CPCP). With the unified framework, we first present two robust trace norm regularized bilateral factorization models for RMC and CPCP problems, which can achieve an orthogonal dictionary and a robust data representation, simultaneously. Then, we apply the alternating direction method of multipliers to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our RBF method compared with the state-of-the-art methods.

Tractable and Scalable Schatten Quasi-Norm Approximations for Rank Minimization

“…The formulations (3), ( 4) and ( 5) can address a wide range of problems, such as MC [13,10], RPCA [2,25,26] (A is the identity operator, and…”

Dynamic Mode Decomposition for Robust PCA with Applications to Foreground/Background Subtraction in Video Streams and Multi-Resolution Analysis

“…From the optimization problem (15.2), we easily find the optimal solution S Ω C = 0 [42,43], where Ω C is the complement of Ω, i.e., the index set of unobserved entries. Consequently, we have the following lemma [42,43].…”

“…Recovering low-rank and sparse matrices from incomplete or even corrupted observations is a common problem in many application areas, including statistics [1,9,51], bioinformatics [37], machine learning [28,47,49,52], computer vision [5,7,42,43,58], and signal and image processing [27,30,38]. In these areas, data often have high dimensionality, such as digital photographs and surveillance videos, which makes inference, learning, and recognition infeasible due to the "curse of dimensionality.…”