Some Observations about Ramanujan Graphs With Applications to Matrix Completion

The matrix completion problem is the following: Suppose a matrix X is unknown except for an upper bound r on its rank. By sampling various entries of X, is it possible to “complete” the remaining entries and recover X exactly? There are two popular approaches to choosing the entries to be sampled. The first approach is to select these entries at random (either with or without replacement). The second approach is to choose the entries to be sampled in a deterministic fashion; this is the approach studied here. We show that if the entries to be sampled correspond to the edges of a so-called Ramanujan graph (defined below), then the measured matrix is in some sense an “optimal” approximation of the unknown matrix. This naturally raises the question of constructing such Ramanujan graphs. It turns out that there are very few explicit constructions of Ramanujan graphs. In this paper, we review the known methods and analyze their suitability for solving the matrix completion problem. The preceding discussion applies to the completion of square matrices. If one is interested in completing rectangular matrices, then it becomes necessary to choose the sample set to correspond to the edges of a Ramanujan bigraph. Until now, there has not been a single explicit construction of a Ramanujan bigraph. Two such constructions are given here. When specialized to the case of square matrices, our construction generates a new family of Ramanujan graphs.