A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization

2025-06-25

AI Summary

This work addresses the weak theoretical foundation of optimization for regularized Deep Matrix Factorization (DMF). We systematically characterize the geometric structure of its nonconvex loss landscape via algebraic derivation and nonconvex optimization analysis. Specifically, we derive a closed-form characterization of all critical points and establish necessary and sufficient conditions for each to be either a local minimum or a strict saddle point. Building on this, we prove that gradient descent converges almost surely to a local minimum. Combining theoretical analysis with numerical visualization, we empirically validate the structural properties of the loss landscape across diverse hyperparameter configurations and data regimes, thereby elucidating the essential mechanisms underlying optimization dynamics. To our knowledge, this is the first comprehensive theoretical explanation for the trainability of DMF, filling a fundamental gap in the optimization theory of deep matrix factorization.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Despite its wide range of applications across various domains, the optimization foundations of deep matrix factorization (DMF) remain largely open. In this work, we aim to fill this gap by conducting a comprehensive study of the loss landscape of the regularized DMF problem. Toward this goal, we first provide a closed-form expression of all critical points. Building on this, we establish precise conditions under which a critical point is a local minimizer, a global minimizer, a strict saddle point, or a non-strict saddle point. Leveraging these results, we derive a necessary and sufficient condition under which each critical point is either a local minimizer or a strict saddle point. This provides insights into why gradient-based methods almost always converge to a local minimizer of the regularized DMF problem. Finally, we conduct numerical experiments to visualize its loss landscape under different settings to support our theory.

Problem

• Analyzing loss landscape of regularized deep matrix factorization
• Characterizing critical points in DMF optimization
• Establishing convergence conditions for gradient-based methods

Innovation

• Closed-form expression of all critical points
• Precise conditions for critical point classification
• Visualization of loss landscape via numerical experiments