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 $λ\leq \{(H-H_0)\min(M,N)+H_0(M+N-H_0)\}/2$. 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 $λ=H_0(M+N-H_0)/2$. 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.
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Cite this article
APA 7
Hayashi, N., Maeda, Y., & Esaki, Y. (2026). Singularities of Non-negative Matrix Factorization and their application to Bayesian inference. https://omanscience.com/en/articles/singularities-of-non-negative-matrix-factorization-and-their-application-to-bayesian-inference
MLA 9
Hayashi, Naoki, et al. "Singularities of Non-negative Matrix Factorization and their application to Bayesian inference." https://omanscience.com/en/articles/singularities-of-non-negative-matrix-factorization-and-their-application-to-bayesian-inference.
Chicago (author–date)
Hayashi, Naoki, Yota Maeda, and Yasushi Esaki. 2026. "Singularities of Non-negative Matrix Factorization and their application to Bayesian inference." https://omanscience.com/en/articles/singularities-of-non-negative-matrix-factorization-and-their-application-to-bayesian-inference.
Harvard
Hayashi, N., Maeda, Y. and Esaki, Y. (2026) 'Singularities of Non-negative Matrix Factorization and their application to Bayesian inference', Available at: https://omanscience.com/en/articles/singularities-of-non-negative-matrix-factorization-and-their-application-to-bayesian-inference.
Vancouver
Hayashi N, Maeda Y, Esaki Y. Singularities of Non-negative Matrix Factorization and their application to Bayesian inference. https://omanscience.com/en/articles/singularities-of-non-negative-matrix-factorization-and-their-application-to-bayesian-inference
IEEE
N. Hayashi, Y. Maeda, and Y. Esaki, "Singularities of Non-negative Matrix Factorization and their application to Bayesian inference," https://omanscience.com/en/articles/singularities-of-non-negative-matrix-factorization-and-their-application-to-bayesian-inference.