Matrix Completion with Nonconvex Regularization: Spectral Operators and Scalable Algorithms

In this paper, we study the popularly dubbed matrix completion problem, where the task is to "fill in" the unobserved entries of a matrix from a small subset of observed entries, under the assumption that the underlying matrix is of low-rank. Our contributions herein, enhance our prior work on nuclear norm regularized problems for matrix completion (Mazumder et al., 2010) by incorporating a continuum of nonconvex penalty functions between the convex nuclear norm and nonconvex rank functions. Inspired by SOFT-IMPUTE (Mazumder et al., 2010; Hastie et al., 2016), we propose NC-IMPUTE- an EM-flavored algorithmic framework for computing a family of nonconvex penalized matrix completion problems with warm-starts. We present a systematic study of the associated spectral thresholding operators, which play an important role in the overall algorithm. We study convergence properties of the algorithm. Using structured low-rank SVD computations, we demonstrate the computational scalability of our proposal for problems up to the Netflix size (approximately, a $500,000 \times 20, 000$ matrix with $10^8$ observed entries). We demonstrate that on a wide range of synthetic and real data instances, our proposed nonconvex regularization framework leads to low-rank solutions with better predictive performance when compared to those obtained from nuclear norm problems. Implementations of algorithms proposed herein, written in the R programming language, are made available on github.