This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent exhibits a low-rank phase in the noiseless setting. These conditions show how the target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. We then analyze the perturbed gradient descent dynamics, proving convergence guarantees and quantifying how the perturbation affects iteration complexity and eigenvalue recovery. Finally, we show that the low-rank phase persists under perturbation, with explicit dependence on the perturbation size. Numerical experiments support the theoretical findings.
We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate I(δ;⋅) whose boundary-inward property induces a monotone state parameter δt, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-critical step sizes, the reduced dynamics exhibit period-2 behavior consistent with edge-of-stability phenomena. We further show that the scalar certificate is not an ad hoc algebraic construction: under structural axioms and a natural state-parameter normalization, it is uniquely determined by the monotonicity mechanism. Numerical experiments suggest that this state-dependent Lyapunov mechanism persists beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.
We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss--Newton (RGN) methods. For an n×n matrix of rank r with incoherence parameter μ and condition number κ, the two methods achieve exact recovery with high probability from O(μnrlognlog(nκ)) and O(μnrlognlog(2μrκ)) observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.
We explore Riemannian optimization techniques for rank-factored matrix parameters, targeting contemporary deep learning applications. We examine ten points in the algorithm design space: two geometries for rank-r matrices, three geometries for rank-r partial isometries, and block-matrix variants of these five, where factors are shared across block-rows and block-columns. We apply our methods to the multihead attention parameters in small language models. After tuning learning rates, our methods do not conclusively outperform an AdamW baseline. Our implementations are available online.