Deterministic Completion of Rectangular Matrices Using Asymmetric Ramanujan Graphs: Exact and Stable Recovery

In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to recover $X$ exactly with noise-free measurements, or to construct a good approximation of $X$ with noisy measurements? Existing solutions to these problems involve sampling the elements uniformly and at random, and can guarantee exact recovery of the unknown matrix only with high probability. In this paper, we present a \textit{deterministic} sampling method for matrix completion. We achieve this by choosing the sampling set as the edge set of an asymmetric Ramanujan bigraph, and constrained nuclear norm minimization is the recovery method. Specifically, we derive sufficient conditions under which the unknown matrix is completed exactly with noise-free measurements, and is approximately completed with noisy measurements, which we call "stable" completion. The conditions derived here are only sufficient and more restrictive than random sampling. To study how close they are to being necessary, we conducted numerical simulations on randomly generated low rank matrices, using the LPS families of Ramanujan graphs. These simulations demonstrate two facts: (i) In order to achieve exact completion, it appears sufficient to choose the degree $d$ of the Ramanujan graph to be $\geq 3r$. (ii) There is a "phase transition," whereby the likelihood of success suddenly drops from 100\% to 0\% if the rank is increased by just one or two beyond a critical value. The phase transition phenomenon is well-known and well-studied in vector recovery using $\ell_1$-norm minimization. However, it is less studied in matrix completion and nuclear norm minimization, and not much understood.