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.

翻译：非凸方法已成为低秩矩阵估计的主流手段，该问题广泛出现在机器学习和人工智能中，用于学习和表示高维数据。现有针对这些方法的分析往往需要额外正则化来缓解非凸性，尽管这种正则化在实践中通常并非必要。此外，多数分析依赖于难以推广至更复杂场景的特定问题论证。本文针对一大类低秩矩阵估计问题，建立了一个研究非凸过程的理论框架。我们并非聚焦于特定模型，而是揭示了一个基本机理，用以解释非凸过程为何能在低秩估计中表现良好。本文的核心工具是一种"良性正则化器"，它不会改变原始更新规则，却能构建算法等价的局部强凸形式。这一视角揭示了非凸过程中隐藏的伪装凸性，并为非凸低秩矩阵估计的理论保证提供了新路径。