No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis

TL;DR

This paper introduces a unified geometric framework for non-convex low-rank matrix problems, showing all local minima are global and no high-order saddle points exist, explaining the success of simple algorithms.

Contribution

It provides a comprehensive analysis that unifies and extends understanding of the optimization landscape for various low-rank matrix problems.

Findings

All local minima are globally optimal.

No high-order saddle points exist.

Simple algorithms like stochastic gradient descent converge globally.

Abstract

In this paper we develop a new framework that captures the common landscape underlying the common non-convex low-rank matrix problems including matrix sensing, matrix completion and robust PCA. In particular, we show for all above problems (including asymmetric cases): 1) all local minima are also globally optimal; 2) no high-order saddle points exists. These results explain why simple algorithms such as stochastic gradient descent have global converge, and efficiently optimize these non-convex objective functions in practice. Our framework connects and simplifies the existing analyses on optimization landscapes for matrix sensing and symmetric matrix completion. The framework naturally leads to new results for asymmetric matrix completion and robust PCA.