  Latent Dirichlet allocation (LDA) is useful in document analysis, image
processing, and many information systems; however, its generalization
performance has been left unknown because it is a singular learning machine to
which regular statistical theory can not be applied.
  Stochastic matrix factorization (SMF) is a restricted matrix factorization in
which matrix factors are stochastic; the column of the matrix is in a simplex.
SMF is being applied to image recognition and text mining. We can understand
SMF as a statistical model by which a stochastic matrix of given data is
represented by a product of two stochastic matrices, whose generalization
performance has also been left unknown because of non-regularity.
  In this paper, by using an algebraic and geometric method, we show the
analytic equivalence of LDA and SMF, both of which have the same real log
canonical threshold (RLCT), resulting in that they asymptotically have the same
Bayesian generalization error and the same log marginal likelihood. Moreover,
we derive the upper bound of the RLCT and prove that it is smaller than the
dimension of the parameter divided by two, hence the Bayesian generalization
errors of them are smaller than those of regular statistical models.
