Convex and Nonconvex Approaches for the Matrix Completion Problem

Missing data is a common occurrence in modern applications such as medical imaging and movie ratings data sets for recommendation systems. Matrix completion is a procedure for recovering the missing entries in a matrix on the basis of the known entries. It is assumed that there is a linear relationship between the rows or columns of the matrix to be recovered, i.e., a small number of rows (or columns) can be used to represent the whole matrix. The matrix completion problem cannot be solved uniquely without imposing constraints such as the low-rank assumption. There are two classes of optimization techniques for sparse matrix completion, namely convex and non-convex approaches. Under the weak assumption that the matrix entries are missing in a non-uniform and deterministic fashion, the matrix completion problem can be formulated as a non-convex bilinear optimization problem that can be solved via the alternating proximal algorithm. On the other hand, the soft-impute algorithm is a convex optimization approach that can deliver fast convergence by exploiting warm-start. In this paper, we test the (convex) soft-impute and (non-convex) alternating proximal matrix completion methods using some real data and furthermore, test [1]’s claim that non-convex approaches perform better than convex approaches. We show a comparison of the two approaches in terms of computational complexity, execution time, and prediction accuracy. In addition, we propose a systematic procedure to perform model selection in both algorithms.