Dimension of Marginals of Kronecker Product Models; Geometry of hidden-visible products of exponential families
Guido Montúfar and Jason Morton
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Submission date: 10. Nov. 2015
published in: SIAM journal on applied algebra and geometry, 1 (2017) 1, p. 126-151
DOI number (of the published article): 10.1137/16M1077489
with the following different title: Dimension of marginals of Kronecker product models
MSC-Numbers: 14T05, 52B05
Keywords and phrases: expected dimension, tropical geometry, secant variety, restricted Boltzmann machine, inference function, kronecker product, Hadamard product, Khatri-Rao product
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A Kronecker product model is the set of visible marginal probability distributions of an exponential family whose sufficient statistics matrix factorizes as a Kronecker product of two matrices, one for the visible variables and one for the hidden variables. We estimate the dimension of these models 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. Additionally, we prove that the binary restricted Boltzmann machine always has the expected dimension.