On the Generic Low-Rank Matrix Completion Problems

Abstract

This paper investigates the low-rank matrix completion (LRMC) problem from a generic view. Unlike most existing work which focused on numerically recovering exact or approximate missing matrix entries from the observed ones, the only available information herein is the pattern (structure) of observed/missing entries, and the observed entries are classified into two types, namely, fixed zero entries and unknown generic entries. The problem is whether there exists a matrix completion with a prescribed rank r for almost all values of the unknown generic entries except for a set of zero measure, which is called the generic low-rank matrix completion (GLRMC) problem. We first justify the existence of genericity in the complex field for this problem, that is, depending on the pattern of observed/missing entries, for almost all values of the unknown generic entries, either the answer to that problem is yes, or the answer is no. We then provide necessary and sufficient conditions for the feasibility of the GLRMC with the constraint that the rank of the completion reduces at least one. Afterward, we provide a sufficient condition and a necessary condition (which we conjecture to be sufficient) for the general case. Based on them, two randomized algorithms are proposed to determine upper and lower bounds for the generic minimum rank of the matrix completions. Our approaches are based on the algebraic geometry theory and the basis preservation principle discovered herein. Finally, numerical experiments are given to corroborate the theoretical findings and the effectiveness of the proposed algorithms.