Global Convergence of Four-Layer Matrix Factorization under Random Initialization

2025-11-13

AI Summary

Deep matrix factorization lacks theoretical guarantees of global convergence under random initialization, particularly for gradient descent optimization. Method: This paper establishes, for the first time, polynomial-time global convergence of four-layer matrix factorization under gradient descent, assuming bounded condition number of the target matrix and standard balanced weight regularization. Leveraging dynamical systems modeling and matrix spectral analysis, we introduce a novel saddle-point escape technique that rigorously characterizes the evolution of singular values across all layers. Contribution/Results: Our work fills a fundamental theoretical gap in deep matrix factorization by proving global convergence—previously unattested for deep linear models—and uncovers an implicit inter-layer coordination mechanism inherent to gradient descent. This reveals how layer-wise updates collectively drive optimization, offering critical theoretical insights into the training dynamics of deep neural networks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Gradient descent dynamics on the deep matrix factorization problem is extensively studied as a simplified theoretical model for deep neural networks. Although the convergence theory for two-layer matrix factorization is well-established, no global convergence guarantee for general deep matrix factorization under random initialization has been established to date. To address this gap, we provide a polynomial-time global convergence guarantee for randomly initialized gradient descent on four-layer matrix factorization, given certain conditions on the target matrix and a standard balanced regularization term. Our analysis employs new techniques to show saddle-avoidance properties of gradient decent dynamics, and extends previous theories to characterize the change in eigenvalues of layer weights.

Problem

• Establishes global convergence for four-layer matrix factorization under random initialization
• Analyzes gradient descent dynamics with saddle-avoidance properties in deep networks
• Extends eigenvalue characterization of layer weights in deep matrix factorization

Innovation

• Global convergence for four-layer matrix factorization
• Polynomial-time guarantee with random initialization
• Saddle-avoidance techniques and eigenvalue analysis