Variational Approximation Error in Bayesian Non-negative Matrix Factorization
This work addresses a theoretical gap in variational Bayesian NMF for researchers in machine learning and statistics, though it is incremental as it builds on existing methods to analyze approximation error.
The paper tackles the problem of quantifying variational approximation error in Bayesian non-negative matrix factorization (NMF), deriving a lower bound for the error that depends on hyperparameters and true nonnegative rank, with numerical experiments validating the theoretical result.
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.