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

arXiv: https://arxiv.org/abs/2605.05446 · HTML: https://mlsanity.com/paper/2605.05446/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 55 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 6 of 6)

- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent](https://mlsanity.com/paper/1605.07051/) (2016)
- [Statistical Inference on Latent Space Models for Network Data](https://mlsanity.com/paper/2312.06605/) (2023)
- [Identifiability and Inference for Generalized Latent Factor Models](https://mlsanity.com/paper/2508.05866/) (2025)
- [Beyond Vintage Rotation: Bias-Free Sparse Representation Learning with Oracle Inference](https://mlsanity.com/paper/2602.22590/) (2026)

## Abstract (arXiv)

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.