Variational Approximation Error in Bayesian Non-negative Matrix Factorization

TL;DR

This paper provides a theoretical analysis of the variational approximation error in Bayesian non-negative matrix factorization (NMF), deriving bounds that quantify how closely variational methods approximate the true Bayesian posterior.

Contribution

It introduces an algebraic geometric approach to analyze the approximation error in VBNMF and derives bounds based on the model's hyperparameters and true rank.

Findings

Derived an upper bound for the learning coefficient in Bayesian NMF.

Established a lower bound for the variational approximation error.

Numerical experiments confirm the theoretical bounds.

Abstract

Non-negative matrix factorization (NMF) is a knowledge discovery method that is used in many fields. Variational inference and Gibbs sampling methods for it are also wellknown. However, the variational approximation error has not been clarified yet, because NMF is not statistically regular and the prior distribution used in variational Bayesian NMF (VBNMF) has zero or divergence points. In this paper, using algebraic geometrical methods, we theoretically analyze the difference in negative log evidence (a.k.a. free energy) between VBNMF and Bayesian NMF, i.e., the Kullback-Leibler divergence between the variational posterior and the true posterior. We derive an upper bound for the learning coefficient (a.k.a. the real log canonical threshold) in Bayesian NMF. By using the upper bound, we find a lower bound for the approximation error, asymptotically. The result quantitatively shows how well the VBNMF algorithm can approximate Bayesian NMF; the lower bound depends on the hyperparameters and the true nonnegative rank. A numerical experiment demonstrates the theoretical result.