Dimension of Marginals of Kronecker Product Models

A Kronecker product model is an exponential family whose sufficient statistics matrix factorizes as a Kronecker product of two matrices, one assigned to a visible set of variables and the other to a hidden set of variables. We estimate the dimension of the set of visible marginal probability distributions by the maximum rank of the Jacobian in the limit of large parameters. The limit is described by the tropical morphism: a piecewise linear map with pieces corresponding to slicings of the visible matrix by the normal fan of the hidden matrix. We obtain combinatorial conditions under which the model has the expected dimension, equal to the minimum of the number of natural parameters and the dimension of the ambient probability simplex. Furthermore, we prove that the binary restricted Boltzmann machine always has the expected dimension.