Low-Rank Matrix Recovery via Continuation-Based Approximate Low-Rank Minimization

Abstract

Low-rank matrix recovery (LRMR) is to recover the underlying low-rank structure from the degraded observations, and has myriad formulations, for example robust principal component analysis (RPCA). As a core component of LRMR, the low-rank approximation model attempts to capture the low-rank structure by approximating the \(\ell _0\)-norm of all the singular values of a low-rank matrix, i.e., the number of the non-zero singular values. Towards this purpose, this paper develops a low-rank approximation model by jointly combining a parameterized hyperbolic tangent (tanh) function with the continuation process. Specificially, the continuation process is exploited to impose the parameterized tanh function to approximate the \(\ell _0\)-norm. We then apply the proposed low-rank model to RPCA and refer to it as tanh-RPCA. Convergence analysis on optimization, and experiments of background subtraction on seven challenging real-world videos show the efficacy of the proposed low-rank model through comparing tanh-RPCA with several state-of-the-art methods.