Stability of Low-Rank Implicit Regularization in Perturbed Deep Matrix Factorization

תקציר מקורי באנגליתarXiv:2605.28613v2 Announce Type: replace-cross Abstract: This paper studies the stability of low-rank implicit regularization in deep matrix factorization, a tractable model for understanding how gradient-based training can favor low-complexity structure. We first revisit the noiseless setting and derive sufficient spectral conditions under which gradient descent exhibits a nonempty low-rank interval. These conditions clarify how the target spectrum, initialization, and step size jointly determine when a low-rank phase is observable along the optimization trajectory. We then analyze the perturbed problem, where the target matrix is subject to an additive perturbation. By studying the perturbed gradient descent dynamics at the eigenvalue level, we prove convergence guarantees and quantify