Singularities of Non-negative Matrix Factorization and their application to Bayesian inference

2026-09-28

AI Summary

This study addresses the Bayesian asymptotic behavior of non-negative matrix factorization (NMF) as a singular statistical model, which is governed by the real log canonical threshold (RLCT). Recognizing the need for precise characterization of local geometric structures, this work leverages singular learning theory and Bayesian asymptotic analysis. By employing local analytic normal form techniques to decouple linear coordinates, we derive and refine upper bound formulas for the RLCT. Specifically, we establish novel RLCT upper bounds that are strictly tighter than existing results for rank three or higher, along with exact values under certain conditions. These findings further determine the dominant coefficients of the expected Bayesian generalization error and the free energy. Consequently, this research provides a rigorous theoretical foundation for Bayesian inference in NMF.

Abstract

Non-negative matrix factorization (NMF) is a singular statistical model whose Bayesian asymptotics are governed by the real log canonical threshold (RLCT). We study the local geometry of the factorization map and derive an upper bound for the RLCT of NMF. Let $H$ be the model inner dimension and $H_0$ the non-negative rank of the true $M\times N$ matrix. Assuming that the true matrix admits a strictly positive factorization of inner dimension $H_0$ in the interior of the parameter domain, we prove, for smooth positive priors, that $\lambda\leq \{(H-H_0)\min(M,N)+H_0(M+N-H_0)\}/2$. This bound strictly improves the previous bound when $H_0\geq3$. The proof uses a local analytic normal form that separates independent linear coordinates from a residual matrix product. When $H=H_0$ also equals the ordinary rank of the true matrix, we obtain the exact value $\lambda=H_0(M+N-H_0)/2$. Under the standard assumptions of singular learning theory, these results bound the leading coefficients of the expected Bayesian generalization error and the Bayesian free energy.

Problem

• Non-negative Matrix Factorization
• Real Log Canonical Threshold
• Bayesian Inference
• Singular Learning Theory
• Generalization Error

Innovation

• Non-negative Matrix Factorization
• Real Log Canonical Threshold
• Bayesian Inference
• Singular Learning Theory
• Local Analytic Normal Form