When Does a Mixture of Products Contain a Product of Mixtures?
Guido Montúfar and Jason Morton
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Submission date: 18. Sep. 2014
published in: SIAM journal on discrete mathematics, 29 (2015) 1, p. 321-347
DOI number (of the published article): 10.1137/140957081
MSC-Numbers: 51M20, 60C05, 68Q32, 14Q15
Keywords and phrases: linear threshold function, Hadamard product, zonotope, Tensor rank, hyperplane arrangement
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We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.