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Tighter upper bound of real log canonical threshold of non-negative matrix factorization and its application to Bayesian inference

Non-negative matrix factorization (NMF) is now used for knowledge discovery in machine learning, however, its mathematical properties have not yet established, because it is statistically singular. Recently, we derived an upper bound of the real log canonical threshold (RLCT) by using an algebraic geometrical method and used it to clarify the theoretical bound of the Bayesian generalization error of NMF. In this paper, we derive the exact value of the RLCT of NMF in the case that the rank of the matrix is not larger than two, and prove the tighter upper bound for general cases.