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

TL;DR

<3-5 sentence high-level summary>The paper introduces a unified geometric framework for nonconvex low-rank problems, showing that matrix sensing, matrix completion, and robust PCA share a landscape with no spurious local minima and strict saddles under appropriate RIP and incoherence conditions. By exploiting a Burer–Monteiro factorization and a single direction of improvement, the authors derive conditions under which all local minima are globally optimal and saddle points are strictly bad, enabling efficient convergence of simple algorithms. The framework also provides a systematic way to handle asymmetry and regularization, reducing asymmetric problems to symmetric PSD ones and enabling polynomial-time runtimes via saddle-avoiding methods. These insights unify prior results, extend them to robust PCA and asymmetric settings, and point toward faster convergence under favorable local geometry.

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.

Theorems & Definitions (100)

• Theorem 1: informal
• Corollary 2: informal
• Definition 1: Optimality Condition
• Definition 2
• Definition 3
• Theorem 3
• Definition 4
• Theorem 4
• Definition 5
• Theorem 5