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

TL;DR

This paper introduces a theoretical framework revealing an inherent convexity in nonconvex low-rank matrix estimation methods, explaining their effectiveness without the need for additional regularization.

Contribution

It uncovers a fundamental mechanism through a benign regularizer that transforms nonconvex procedures into a locally strongly convex form, broadening theoretical understanding.

Findings

Identifies a disguised convexity in nonconvex low-rank estimation methods.

Provides a general theoretical framework applicable across various models.

Shows that benign regularizers can explain the success of nonconvex algorithms.

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.