Geometry of the restricted Boltzmann machine
The restricted Boltzmann machine is a graphical model for binary
random variables. Based on a complete bipartite graph separating
hidden and observed variables, it is the binary analog to the factor
analysis model. We study this graphical model from the perspectives of
algebraic statistics and tropical geometry, starting with the
observation that its Zariski closure is a Hadamard power of the first
secant variety of the Segre variety of projective lines. We derive a
dimension formula for the tropicalized model, and we use it to show
that the restricted Boltzmann machine is identifiable in many cases.
Our methods include coding theory and geometry of linear threshold
functions.