Message-Passing Algorithms and Homology
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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 u̇ = δφ. 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_α) ∈ A_0, for α running over the factor nodes of a given region graph. The boundary operator δ mapping heat fluxes (φ_αβ) ∈ A_1 to a subspace δ A_1 ⊆ A_0 is the discrete analog of a divergence. The total energy H = ∑_α u_α defining the global probability p = e^-H / Z is in one-to-one correspondence with a homology class [u] = u + δ A_1 of interaction potentials, so that total energy remains constant when u evolves up to a boundary term δφ. 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
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