Deterministic Completion of Rectangular Matrices Using Ramanujan Bigraphs -- I: Error Bounds and Exact Recovery

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 $m \ll n_r n_c$ of the 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 the exact recovery of the unknown matrix, but only with high probability. At present there are very few deterministic methods, and they mostly apply only to square matrices. The focus in the present paper is on deterministic methods that work for rectangular as well as square matrices, and where possible, can guarantee exact recovery of the unknown matrix. We achieve this by choosing the elements to be sampled as the edge set of an asymmetric Ramanujan graph or Ramanujan bigraph. For such a measurement matrix, we (i) derive bounds on the error between a scaled version of the sampled matrix and unknown matrix; (ii) derive bounds on the recovery error when max norm minimization is used, and (iii) present suitable conditions under which the unknown matrix can be recovered exactly via nuclear norm minimization. In the process we streamline some existing proofs and improve upon them, and also make the results applicable to rectangular matrices. This raises two questions: (i) How can Ramanujan bigraphs be constructed? (ii) How close are the sufficient conditions derived in this paper to being necessary? Both questions are studied in a companion paper.