Geometric View of Restricted Boltzmann Machines

Restricted Boltzmann Machines (RBMs) are fundamental probabilistic models in machine learning, defined through a bipartite network of visible and hidden units. Despite their conceptual simplicity, RBMs reveal a rich interplay between combinatorics, geometry, and statistical learning theory. I will present a mathematical view of RBMs, emphasizing their structure as exponential families and as Hadamard products of simpler models. We will examine how the architecture constrains the set of realizable probability distributions and approximation errors, highlighting associated polytopes and varieties. I will also discuss connections to tropical geometry, tensor factorizations, mixture models, and ReLU neural networks.