Singularities of Non-negative Matrix Factorization and their application to Bayesian inference
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 H0 the non-negative rank of the true M× N matrix. Assuming that the true matrix admits a strictly positive factorization of inner dimension H0 in the interior of the parameter domain, we prove, for smooth positive priors, that λ≤ {(H−H0)min(M,N)+H0(M+N−H0)}/2. This bound strictly improves the previous bound when H0≥3. The proof uses a local analytic normal form that separates independent linear coordinates from a residual matrix product. When H=H0 also equals the ordinary rank of the true matrix, we obtain the exact value λ=H0(M+N−H0)/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.
Related
- Kumaraswamy-Teissier Distribution: Bayesian and Non-Bayesian Inference, Regression Modeling and Climate Application
- Matrix factorization of quotient singularities of type A and D
- A hybrid approach combining a phylogenetic method and Approximate Bayesian Computation Random Forest for phylogenetic network inference: application to the rice domestication process in Asia
- Non-existence of Negative Weight Derivations of the Moduli Algebras of Isolated Weighted Homogeneous Hypersurface Singularities
- Gibbs Sampling for Bayesian Generalized Poisson Matrix Factorization
BibTeX
@misc{hayashi2026singularities,
title = {{Singularities of Non-negative Matrix Factorization and their application to Bayesian inference}},
author = {Naoki Hayashi and Yota Maeda and Yasushi Esaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34043},
eprint = {2609.34043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34043},
}
RIS
TY - UNPB
TI - Singularities of Non-negative Matrix Factorization and their application to Bayesian inference
AU - Naoki Hayashi
AU - Yota Maeda
AU - Yasushi Esaki
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34043
UR - https://arxiv.org/abs/2609.34043
AB - 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× 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 λ≤ {(H−H_0)min(M,N)+H_0(M+N−H_0)}/2. This bound strictly improves the previous bound when H_0≥3. 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 λ=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.
ER -