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

arXiv:2609.34043v1 Announce Type: cross 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 Mimes 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 lambdaleq {(H-H_0)min(M,N)+H_0(M+N-H_0)}/2. This bound strictly improves the previous bound when H_0geq3. 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.

Source: arXiv cs.LG | 2026-09-29