# Over-parameterization Is Neither Robust nor Fragile in Matrix Sensing: A Geometric Theory of Solution Robustness

arXiv: https://arxiv.org/abs/2610.18318 · HTML: https://mlsanity.com/paper/2610.18318/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 37 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Builds on (most cited first, 17 of 17)

- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)
- [Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent](https://mlsanity.com/paper/2005.08898/) (2020)
- [The non-convex Burer-Monteiro approach works on smooth semidefinite programs](https://mlsanity.com/paper/1606.04970/) (2016)
- [The Power of Preconditioning in Overparameterized Low-Rank Matrix Sensing](https://mlsanity.com/paper/2302.01186/) (2023)
- [Low-Rank Matrix Recovery with Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number](https://mlsanity.com/paper/2010.13364/) (2020)
- [Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery](https://mlsanity.com/paper/1901.01631/) (2019)
- [The Global Optimization Geometry of Low-Rank Matrix Optimization](https://mlsanity.com/paper/1703.01256/) (2017)
- [Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery](https://mlsanity.com/paper/2109.11154/) (2021)
- [Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs](https://mlsanity.com/paper/1804.02008/) (2018)
- [Noisy Low-rank Matrix Optimization: Geometry of Local Minima and Convergence Rate](https://mlsanity.com/paper/2203.03899/) (2022)
- [Sharp Global Guarantees for Nonconvex Low-rank Recovery in the Noisy Overparameterized Regime](https://mlsanity.com/paper/2104.10790/) (2021)
- [Nonconvex Factorization and Manifold Formulations are Almost Equivalent in Low-rank Matrix Optimization](https://mlsanity.com/paper/2108.01772/) (2021)

## Abstract (arXiv)

Over-parameterization is ubiquitous in modern machine learning, yet its effect on robustness remains poorly understood. We study this question in matrix sensing, a canonical non-convex problem that can also be viewed, under an appropriate parameterization, as training a two-layer quadratic neural network. We develop a rank-aware robustness theory for the Burer-Monteiro factorization when the search rank k may exceed the target rank r_star, thereby capturing over-parameterization. Our analysis shows that robustness depends jointly on the degree of over-parameterization and the local geometry of the attained second-order stationary point: increasing k generally worsens sensitivity, but stationary points with the same search rank can exhibit substantially different robustness. Motivated by this observation, we propose a trajectory-aware algorithm that uses spectral balancing and randomized sketching to steer optimization toward less sensitive solutions. In matrix-sensing experiments, the method reduces paired sensitivity by 18-29 percent for isotropic operators and 87-89 percent for weak-tail operators, without degrading reconstruction accuracy. These results demonstrate that parameter count alone does not determine robustness and identify solution geometry and the optimization trajectory as important levers for improving it.