Exact Completion of Rectangular Matrices Using Ramanujan Bigraphs

Abstract

In this paper, we study the matrix completion problem: Suppose $X \in {\mathbb{R}^{{n_r} \times {n_c}}}$ is unknown except for an upper bound r on its rank. By measuring a small number m ≪ nrnc of elements of X, is it possible to recover X exactly, or at least, to construct a reasonable approximation of X? At present, there are two approaches to choosing the sample set, namely probabilistic and deterministic. Probabilistic methods can guarantee exact recovery of the unknown matrix, but only with high probability. In this approach, samples are taken uniformly at random. Therefore we need to start sampling for every new matrix afresh. In the deterministic approach, sampling points can be kept fixed. At present, there are very few deterministic methods, and they mostly apply only to square matrices. In this paper, we present a deterministic method for selecting the sample set that can guarantee the exact recovery of the unknown matrix. This approach works for the recovery of rectangular as well as square matrices. We achieve this by choosing the elements to be sampled as the edge set of a Ramanujan bigraph. If samples are the edge set of a Ramanujan bigraph, then we can recover the unknown matrix from that sample set using nuclear norm minimization. A companion paper discusses the explicit construction of Ramanujan bigraphs. We provide a sufficient condition, that is if the samples taken are of the order of r3 then we can recover the unknown entries exactly if the unknown matrix satisfies some coherence condition. We believe this the first sufficient condition available using deterministic sampling technique and nuclear norm minimization.