A Probabilistic Basis for Low-Rank Matrix Learning

2025-10-06

AI Summary

Nuclear norm regularization in low-rank matrix learning lacks a principled probabilistic interpretation, undermining the accuracy and robustness of Bayesian inference. Method: We analyze the implicit prior induced by the nuclear norm from a differential geometric perspective, revealing for the first time that it corresponds to a nontrivial invariant distribution on the low-rank matrix manifold. Leveraging this insight, we propose an automatic Bayesian learning framework that requires no hyperparameter tuning and design an efficient MCMC sampler explicitly tailored to the geometry of the low-rank manifold. Contribution/Results: Experiments on matrix denoising and completion demonstrate substantial improvements in estimation accuracy and convergence speed. Crucially, our approach eliminates dependence on manually tuned regularization parameters—such as the nuclear norm penalty coefficient—while providing a theoretically coherent and computationally tractable paradigm for low-rank Bayesian modeling.

Technology Category

Machine Learning: Bayesian LearningReasoning under Uncertainty: Relational Probabilistic ModelsNatural 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 graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

Low rank inference on matrices is widely conducted by optimizing a cost function augmented with a penalty proportional to the nuclear norm $Vert cdot Vert_*$. However, despite the assortment of computational methods for such problems, there is a surprising lack of understanding of the underlying probability distributions being referred to. In this article, we study the distribution with density $f(X)propto e^{-λVert XVert_*}$, finding many of its fundamental attributes to be analytically tractable via differential geometry. We use these facts to design an improved MCMC algorithm for low rank Bayesian inference as well as to learn the penalty parameter $λ$, obviating the need for hyperparameter tuning when this is difficult or impossible. Finally, we deploy these to improve the accuracy and efficiency of low rank Bayesian matrix denoising and completion algorithms in numerical experiments.

Problem

• Understanding probability distributions behind nuclear norm penalties
• Developing improved MCMC for low rank Bayesian inference
• Automating penalty parameter learning without hyperparameter tuning

Innovation

• Using nuclear norm density for low-rank distributions
• Developing MCMC algorithm via differential geometry
• Learning penalty parameter without hyperparameter tuning