A gradual rank increasing process for matrix completion

Abstract

The matrix completion problem is easy to state: let A be a given data matrix in which some entries are unknown. Then, it is needed to assign “appropriate values” to these entries. A common way to solve this problem is to compute a rank-k matrix, B _k , that approximates A in a least squares sense. Then, the unknown entries in A attain the values of the corresponding entries in B _k . This raises the question of how to determine a suitable matrix rank. The method proposed in this paper attempts to answer this question. It builds a finite sequence of matrices $B_{k}, k = 1, 2, \dots$, where B _k is a rank-k matrix that approximates A in a least squares sense. The computational effort is reduced by using B _k-1 as starting point in the computation of B _k . The ability of B _k to serve as substitute for A is measured with two objective functions: a “training” function that measures the distance between the known part of A and the corresponding part of B _k , and a “probe” function that assesses the quality of the imputed entries. Watching the changes in these functions as k increases enables us to find an optimal matrix rank. Numerical experiments illustrate the usefulness of the proposed approach.