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

• URL: http://arxiv.org/abs/2605.05446v1
• Date: Wed, 06 May 2026 21:18:19 GMT
• Title: Convexity in Disguise: A Theoretical Framework for Nonconvex Low-Rank Matrix Estimation
• Abstract summary: Non procedures have emerged as a dominant approach for low-rank matrix estimation.<n>We develop a theoretical framework for non procedures across a class of low-rank matrix estimation problems.
• Score: 2.871875172569441
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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