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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 well-known. 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 non-negative rank. A numerical experiment demonstrates the theoretical result.
This article was published in the following journal.
Name: Neural networks : the official journal of the International Neural Network Society
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A secreted matrix metalloproteinase that is the predominant proteolytic activity in the enamel matrix. The enzyme has a high specificity for dental enamel matrix protein AMELOGENIN.
A transmembrane domain-containing matrix metalloproteinase. It is synthesized as an inactive zymogen that is activated by the proteolytic action of PROPROTEIN CONVERTASES. Matrix metalloproteinase 16 plays a direct role in the cleavage of proteins in the pericellular environment. In addition it can function indirectly by enzymatically activating the proprotein form of other MATRIX METALLOPROTEINASES such as the zymogen of MATRIX METALLOPROTEINASE 2.
A secreted matrix metalloproteinase that is believed to play a role in EXTRACELLULAR MATRIX remodeling and cell fate determination during normal and pathological processes. Matrix metalloproteinase 11 was originally isolated in primary BREAST NEOPLASMS and may be involved in the process of tumorigenesis.
A transmembrane domain-containing matrix metalloproteinase. It is synthesized as an inactive zymogen that is activated by the action of PROPROTEIN CONVERTASES such as FURIN. Matrix metalloproteinase 14 plays a direct role in the cleavage of proteins in the pericellular environment. In addition it can function indirectly by enzymatically activating the proprotein form of MATRIX METALLOPROTEINASE 15.
The introduction of error due to systematic differences in the characteristics between those selected and those not selected for a given study. In sampling bias, error is the result of failure to ensure that all members of the reference population have a known chance of selection in the sample.