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

The paper investigates how over‑parameterization affects robustness in matrix sensing, a non‑convex problem equivalent to training a two‑layer quadratic neural network. It introduces a rank‑aware robustness theory for the Burer‑Monteiro factorization, showing that robustness depends on both the degree of over‑parameterization and the local geometry of second‑order stationary points. Based on these insights, the authors propose a trajectory‑aware algorithm that steers optimization toward less sensitive solutions, achieving significant reductions in paired sensitivity in experiments without harming reconstruction accuracy.

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