Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent

A growing body of recent research is shedding new light on the role of nonconvex optimization for tackling large scale problems in machine learning, signal processing, and convex programming. This work is developing techniques that help to explain the surprising effectiveness of relatively simple first-order algorithms for certain nonconvex optimizations. When applied to problems that can be formulated as semidefinite programs, these techniques can often be viewed as part of a framework proposed by Burer and Monteiro [4]. The Burer-Monteiro technique is based on factoring the semidefinite variable, and applying classical optimization techniques to the resulting nonconvex objective over the factor. While worst-case complexity considerations imply that such an approach cannot succeed in general, a series of recent papers [11, 40, 35, 13, 1] has shown the strategy to be remarkably effective for a number of problems of practical interest, with analytical convergence guarantees and strong empirical performance. In this paper, we enlarge the collection of problems to which the Burer-Monteiro technique can be successfully applied, by analyzing the convergence properties of gradient descent applied to the problem of rectangular matrix completion from incomplete measurements.