Boltzmann machines and hierarchical models

The restricted Boltzmann machine (RBM) is a key statistical model used in deep learning. They are special form of Boltzmann machines where the underlying graph is a bipartite graph. Personally, I am more interested in Boltzmann machines because they represent a class of discrete energy models where the energy is quadratic. The dynamics of the model bears a lot of resemblance to those of Hopfield networks and Ising models. As an aside, normal distributions are continuous energy models where the energy is quadratic and positive definite.

If the energy of the model is a polynomial of higher degree (e.g. cubic, quartic), then the model is hierarchical. They are a kind of graphical model where the underlying graph is a simplicial complex (a special type of hypergraph). Here are some slides and papers on hierachical models:

1. Hierarchical models and monomial ideals
   Daniel Bruynooghe and Henry Wynn

2. Toric ideals in algebraic statistics
   Seth Sullivant

3. Decomposable models
   Franziska Hinkelmann

4. Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees
   Sonja Petrović and Erik Stokes

5. Hierarchical models for contingency tables from the viewpoint of abstract simplicial complex
   Akimichi Takemura

Studying model asymptotics with singular learning theory   Statistics and machine learning