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Variational approximation error in non-negative matrix factorization.

Naoki Hayashi1

  • 1Simulation & Mining Division, NTT DATA Mathematical Systems Inc., 1F Shinanomachi Rengakan, 35, Shinanomachi, Shinjuku-ku, Tokyo, 160-0016, Japan; Department of Mathematical and Computing Science, Tokyo Institute of Technology, Mail-Box W8-42, 2-12-1, Oookayama, Meguro-ku, Tokyo, 152-8552, Japan.

Neural Networks : the Official Journal of the International Neural Network Society
|March 23, 2020
PubMed
Summary

This study clarifies variational approximation error in Variational Bayesian Non-negative Matrix Factorization (VBNMF). Algebraic geometry reveals a lower bound for VBNMF

Keywords:
Bayesian inferenceLearning coefficientNon-negative matrix factorization (NMF)Real log canonical threshold (RLCT)Variational Bayesian methodVariational inference

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Area of Science:

  • Machine Learning
  • Statistical Inference
  • Algebraic Geometry

Background:

  • Non-negative Matrix Factorization (NMF) is a key knowledge discovery technique.
  • Variational inference and Gibbs sampling are common NMF methods.
  • The variational approximation error in Bayesian NMF remains unclear due to statistical irregularities.

Purpose of the Study:

  • Theoretically analyze the approximation error in Variational Bayesian Non-negative Matrix Factorization (VBNMF).
  • Quantify the Kullback-Leibler divergence between variational and true posteriors in VBNMF.
  • Establish a lower bound for the approximation error in VBNMF.

Main Methods:

  • Employ algebraic geometrical methods for theoretical analysis.
  • Derive an upper bound for the learning coefficient in Bayesian NMF.
  • Utilize the derived upper bound to establish an asymptotic lower bound for approximation error.

Main Results:

  • The study provides a theoretical analysis of variational approximation error in VBNMF.
  • An upper bound for the learning coefficient (real log canonical threshold) in Bayesian NMF is derived.
  • An asymptotic lower bound for the approximation error in VBNMF is established, dependent on hyperparameters and non-negative rank.

Conclusions:

  • The derived lower bound quantitatively assesses VBNMF's approximation accuracy.
  • The findings elucidate the relationship between VBNMF approximation error, hyperparameters, and non-negative rank.
  • Numerical experiments validate the theoretical results on VBNMF approximation error.