A Unified Complexity Metric for Nonconvex Matrix Completion and Matrix Sensing in the Rank-one Case

Abstract

In this work, we develop a new complexity metric for an important class of low-rank matrix optimization problems, where the metric aims to quantify the complexity of the nonconvex optimization landscape of each problem and the success of local search methods in solving the problem. The existing literature has focused on two complexity measures. The RIP constant is commonly used to characterize the complexity of matrix sensing problems. On the other hand, the sampling rate and the incoherence are used when analyzing matrix completion problems. The proposed complexity metric has the potential to unify these two notions and also applies to a much larger class of problems. To mathematically study the properties of this metric, we focus on the rank-$1$ generalized matrix completion problem and illustrate the usefulness of the new complexity metric from three aspects. First, we show that instances with the RIP condition have a small complexity. Similarly, if the instance obeys the Bernoulli sampling model, the complexity metric will take a small value with high probability. Moreover, for a one-parameter class of instances, the complexity metric shows consistent behavior to the first two scenarios. Furthermore, we establish theoretical results to provide sufficient conditions and necessary conditions on the existence of spurious solutions in terms of the proposed complexity metric. This contrasts with the RIP and incoherence notions that fail to provide any necessary condition