Robust Matrix Completion Based on Factorization and Truncated-Quadratic Loss Function

Abstract

Robust matrix completion refers to recovering a low-rank matrix given a subset of the entries corrupted by gross errors, and has various applications since many real-world signals can be modeled as low-rank matrices. Most of the existing methods only perform well for noise-free data or those with zero-mean white Gaussian noise, and their performance will be degraded in the presence of outliers. In this paper, based on the factorization framework, we propose a novel robust matrix completion scheme via using the truncated-quadratic loss function, which is non-convex and non-smooth, and half-quadratic theory is adopted for its optimization. By introducing an auxiliary variable, half-quadratic optimization (HO) can transform the loss function into two tractable forms, that is, additive and multiplicative formulations. Block coordinate descent method is then exploited as their solver. Compared with the additive form, the multiplicative variant has lower computational cost since we attempt to take the observations contaminated by outliers as missing entries. Numerical simulations and experimental results based on image inpainting and hyperspectral image recovery demonstrate that our algorithms are superior to the state-of-the-art methods in terms of restoration accuracy and runtime. MATLAB code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/bestzywang</uri> .