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In this paper, we study the popularly dubbed matrix completion problem, wherethe task is to "fill in" the unobserved entries of a matrix from a small subsetof observed entries, under the assumption that the underlying matrix is oflow-rank. Our contributions herein, enhance our prior work on nuclear normregularized problems for matrix completion (Mazumder et al., 2010) byincorporating a continuum of nonconvex penalty functions between the convexnuclear norm and nonconvex rank functions. Inspired by SOFT-IMPUTE (Mazumder etal., 2010; Hastie et al., 2016), we propose NC-IMPUTE- an EM-flavoredalgorithmic framework for computing a family of nonconvex penalized matrixcompletion problems with warm-starts. We present a systematic study of theassociated spectral thresholding operators, which play an important role in theoverall algorithm. We study convergence properties of the algorithm. Usingstructured low-rank SVD computations, we demonstrate the computationalscalability of our proposal for problems up to the Netflix size (approximately,a $500,000 \times 20, 000$ matrix with $10^8$ observed entries). We demonstratethat on a wide range of synthetic and real data instances, our proposednonconvex regularization framework leads to low-rank solutions with betterpredictive performance when compared to those obtained from nuclear normproblems. Implementations of algorithms proposed herein, written in the Rprogramming language, are made available on github.

Matrix Completion with Nonconvex Regularization: Spectral Operators and Scalable Algorithms