Convexity in Disguise: A Theoretical Framework for Nonconvex Low-Rank Matrix Estimation

2026-05-06

AI Summary

This work addresses the long-standing lack of a unified theoretical framework for nonconvex low-rank matrix estimation, where existing approaches often rely on ad hoc regularization and exhibit limited generalizability. The authors propose a general analytical framework that introduces benign regularizers—ones that do not alter the original update rules—to recast nonconvex algorithms into locally strongly convex equivalents, thereby uncovering an intrinsic “disguised convexity.” Remarkably, this framework provides unified theoretical guarantees for a broad class of nonconvex low-rank matrix estimation algorithms without requiring explicit regularization. It explains the empirical observation that such methods often converge efficiently even in the absence of added regularizers and establishes generalizable bounds on both convergence rates and statistical error.

Abstract

Nonconvex methods have emerged as a dominant approach for low-rank matrix estimation, a problem that arises widely in machine learning and AI for learning and representing high-dimensional data. Existing analyses for these methods often require additional regularization to mitigate nonconvexity, even though such regularization is often unnecessary in practice. Moreover, most analyses rely on problem-specific arguments that are difficult to generalize to more complex settings. In this paper, we develop a theoretical framework for studying nonconvex procedures across a broad class of low-rank matrix estimation problems. Rather than focusing on a specific model, we reveal a fundamental mechanism that explains why nonconvex procedures can behave well in low-rank estimation. Our key device is a {\it benign regularizer} that does not alter the original update rule, but yields an equivalent locally strongly convex formulation of the algorithm. This perspective uncovers a disguised convexity inherent in the nonconvex procedure and provides a new route to theoretical guarantees for nonconvex low-rank matrix estimation.

Problem

• nonconvex optimization
• low-rank matrix estimation
• theoretical analysis
• convexity
• regularization

Innovation

• nonconvex optimization
• low-rank matrix estimation
• benign regularizer
• disguised convexity
• theoretical framework