Guaranteed Matrix Completion via Non-convex Factorization

TL;DR

This paper provides theoretical guarantees for non-convex matrix factorization methods in matrix completion, showing convergence to the true low-rank matrix without resampling, and analyzing the local geometry of the optimization landscape.

Contribution

It establishes conditions under which standard algorithms reliably recover the low-rank matrix, advancing understanding of non-convex matrix completion without resampling.

Findings

Algorithms converge to the global optimum under certain conditions.

Any stationary point in a local region is globally optimal.

No resampling needed in the analysis or algorithm.

Abstract

Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity caused by the factorization model, there is a limited theoretical understanding of this formulation. In this paper, we establish a theoretical guarantee for the factorization formulation to correctly recover the underlying low-rank matrix. In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms converge to the global optima of a factorization formulation, and recover the true low-rank matrix. We study the local geometry of a properly regularized factorization formulation and prove that any stationary point in a certain local region is globally optimal. A major difference of our work from the existing results is that we do not need resampling in either the algorithm or its analysis. Compared to other works on nonconvex optimization, one extra difficulty lies in analyzing nonconvex constrained optimization when the constraint (or the corresponding regularizer) is not "consistent" with the gradient direction. One technical contribution is the perturbation analysis for non-symmetric matrix factorization.