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Multivalued Restricted Boltzmann Machines (Hadamard products of mixtures of Segre products), part II

  • Guido Montúfar (Penn State University, USA)
A3 02 (Seminar room)

Abstract

A Restricted Boltzmann Machine (RBM) is a stochastic network with undirected bipartite interactions between a set of observed variables and a set of hidden variables. We are interested in the geometry of the set of all possible marginal probability distributions of the visible variables. In the first lecture I give a comprehensible survey on mathematical aspects of standard RBMs with binary variables.

In the second lecture I discuss RBMs with discrete, finite valued variables. The multivalued RBM hierarchy of models embraces all mixture models of discrete product distributions as well as the standard binary RBMs, and provides a natural view of semi-restricted Boltzmann machines.

From the perspective of algebraic geometry, these models are Hadamard products of secant varieties of Segre embeddings of products of projective spaces. I present a geometric picture of the 'inference functions' and 'distributed mixtures-of-products' represented by multivalued RBMs, and show results on universal approximation, approximation errors, tropicalization and dimension. Hereby extending previous work on the binary case.

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail