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

2026-07-23 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, long

Summary

This paper investigates the stability of low-rank implicit regularization in deep matrix factorization when the target matrix is corrupted by noise. It first establishes sufficient spectral conditions for gradient descent to exhibit a low-rank phase in noiseless settings, detailing how target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. The analysis then extends to perturbed gradient descent dynamics, providing convergence guarantees and quantifying how noise affects iteration complexity and eigenvalue recovery. Crucially, the study demonstrates that the low-rank phase persists under perturbation, with its duration explicitly dependent on the noise size. Numerical experiments, including varying step size and leading eigenvalues, validate these theoretical findings, showing how noise increases approximation error and shifts low-rank plateaus.

Key takeaway

For AI Scientists optimizing deep matrix factorization models with potentially noisy data, this research highlights the importance of understanding spectral properties. Your choice of step size and initialization directly impacts the observability and stability of low-rank implicit regularization. Be aware that noise can shift the effective low-rank approximation intervals and increase approximation error, necessitating careful parameter tuning to maintain desired model performance.

Key insights

Low-rank implicit regularization in deep matrix factorization is robust to bounded additive noise.

Principles

• Spectral conditions govern low-rank phase existence.
• Low-rank implicit regularization persists under perturbation.
• Step size and initialization influence low-rank behavior.

Method

Analyzes perturbed gradient descent dynamics via eigenvalue-level spectral analysis, leveraging the Davis-Kahan theorem for eigenvector perturbation bounds.

In practice

• Consider noise magnitude when setting gradient descent step sizes.
• Evaluate initialization strategies based on target matrix spectrum.
• Expect shifted low-rank approximation plateaus with increased noise.

Topics

• Implicit Regularization
• Deep Matrix Factorization
• Gradient Descent
• Low-Rank Approximation
• Spectral Stability
• Nonconvex Optimization

Best for: Research Scientist, AI Scientist, AI Student

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.