A Geometric Approach to Low-Rank Matrix Completion

TL;DR

This paper introduces a geometric approach to low-rank matrix completion that guarantees performance without requiring matrix incoherence or large size, addressing limitations of traditional Frobenius-based methods.

Contribution

It proposes a new geometric optimization procedure with strong performance guarantees, overcoming issues of discontinuity and non-closed solution sets in existing methods.

Findings

The geometric objective function is continuous everywhere.

The solution set is the closure of the Frobenius-based solution set.

No local minimizers exist in certain scenarios, ensuring reliable convergence.

Abstract

The low-rank matrix completion problem can be succinctly stated as follows: given a subset of the entries of a matrix, find a low-rank matrix consistent with the observations. While several low-complexity algorithms for matrix completion have been proposed so far, it remains an open problem to devise search procedures with provable performance guarantees for a broad class of matrix models. The standard approach to the problem, which involves the minimization of an objective function defined using the Frobenius metric, has inherent difficulties: the objective function is not continuous and the solution set is not closed. To address this problem, we consider an optimization procedure that searches for a column (or row) space that is geometrically consistent with the partial observations. The geometric objective function is continuous everywhere and the solution set is the closure of the solution set of the Frobenius metric. We also preclude the existence of local minimizers, and hence establish strong performance guarantees, for special completion scenarios, which do not require matrix incoherence or large matrix size.