The Global Geometry of Centralized and Distributed Low-rank Matrix Recovery without Regularization

Abstract

Low-rank matrix recovery is a fundamental problem in signal processing and machine learning. A recent very popular approach to recovering a low-rank matrix X is to factorize it as a product of two smaller matrices, i.e., X = UV^T, and then optimize over U, V instead of X. Despite the resulting non-convexity, recent results have shown that many factorized objective functions actually have benign global geometry---with no spurious local minima and satisfying the so-called strict saddle property---ensuring convergence to a global minimum for many local-search algorithms. Such results hold whenever the original objective function is restricted strongly convex and smooth. However, most of these results actually consider a modified cost function that includes a balancing regularizer. While useful for deriving theory, this balancing regularizer does not appear to be necessary in practice. In this work, we close this theory-practice gap by proving that the unaltered factorized non-convex problem, without the balancing regularizer, also has similar benign global geometry. Moreover, we also extend our theoretical results to the field of distributed optimization.

References in corpus (4)

• Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
• How to Escape Saddle Points Efficiently
• Provable Bregman-divergence based Methods for Nonconvex and Non-Lipschitz Problems
• The Landscape of Non-convex Empirical Risk with Degenerate Population Risk

Cited by in corpus (4)

• Beyond Procrustes: Balancing-Free Gradient Descent for Asymmetric Low-Rank Matrix Sensing
• Global convergence of the gradient method for functions definable in o-minimal structures
• Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization
• Landscape Correspondence of Empirical and Population Risks in the Eigendecomposition Problem