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.

翻译：本文研究受扰深度矩阵分解中低秩隐式正则化的稳定性，其中目标矩阵被噪声矩阵污染。我们首先推导出充分的谱条件，在这些条件下无噪声场景中的梯度下降展现出低秩相位。这些条件揭示了目标谱、初始化和步长如何共同决定非空低秩区间的存在性。随后我们分析受扰梯度下降动力学，证明收敛性保证并量化扰动如何影响迭代复杂度与特征值恢复。最后证明低秩相位在扰动下依然存在，且其与扰动大小具有显式依赖关系。数值实验支持了理论发现。