Outlier-Robust Matrix Completion via $\ell _p$ -Minimization

Abstract

Matrix completion refers to recovering a low-rank matrix from only a subset of its possibly noisy entries, and has a variety of important applications because many real-world signals can be modeled by a $n_1 \times n_2$ matrix with rank $r \ll \min (n_1, n_2)$ . Most existing techniques for matrix completion assume Gaussian noise and, thus, they are not robust to outliers. In this paper, we devise two algorithms for robust matrix completion based on low-rank matrix factorization and $\ell _p$ -norm minimization of the fitting error with $0 . The first method tackles the low-rank matrix factorization with missing data by iteratively solving $(n_1+n_2)$ linear $\ell _p$ -regression problems, whereas the second applies the alternating direction method of multipliers (ADMM) in the $\ell _p$ -space. At each iteration of the ADMM, it requires performing a least squares (LS) matrix factorization and calculating the proximity operator of the $p$ th power of the $\ell _p$ -norm. The LS factorization is efficiently solved using linear LS regression while the proximity operator has closed-form solution for $p=1$ or can be obtained by root finding of a scalar nonlinear equation for other values of $p$ . The two proposed algorithms have comparable recovery capability and computational complexity of $\mathcal{O}(K|\Omega |r^2)$ , where $|\Omega |$ is the number of observed entries and $K$ is a fixed constant of several hundreds to thousands and dimension independent. It is demonstrated that they are superior to the singular value thresholding, singular value projection, and alternating projection schemes in terms of computational simplicity, statistical accuracy, and outlier-robustness.