Message-Passing Algorithms and Homology

Abstract: This PhD thesis lays out algebraic and topological structures relevant for the study of probabilistic graphical models. Marginal estimation algorithms are introduced as diffusion equations of the form $\dot u = \delta \varphi$. They generalise the traditional belief propagation (BP) algorithm, and provide an alternative for contrastive divergence (CD) or Markov chain Monte Carlo (MCMC) algorithms, typically involved in estimating a free energy functional and its gradient w.r.t. model parameters. We propose a new homological picture where parameters are a collections of local interaction potentials $(u_\alpha) \in A_0$, for $\alpha$ running over the factor nodes of a given region graph. The boundary operator $\delta$ mapping heat fluxes $(\varphi_{\alpha\beta}) \in A_1$ to a subspace $\delta A_1 \subseteq A_0$ is the discrete analog of a divergence. The total energy $H = \sum_\alpha u_\alpha$ defining the global probability $p = e^{-H} / Z$ is in one-to-one correspondence with a homology class $[u] = u + \delta A_1$ of interaction potentials, so that total energy remains constant when $u$ evolves up to a boundary term $\delta \varphi$. Stationary states of diffusion are shown to lie at the intersection of a homology class of potentials with a non-linear constraint surface enforcing consistency of the local marginals estimates. This picture allows us to precise and complete a proof on the correspondence between stationary states of BP and critical points of a local free energy functional (obtained by Bethe-Kikuchi approximations) and to extend the uniqueness result for acyclic graphs (i.e. trees) to a wider class of hypergraphs. In general, bifurcations of equilibria are related to the spectral singularities of a local diffusion operator, yielding new explicit examples of the degeneracy phenomenon. Work supervised by Pr. Daniel Bennequin