{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/message-passing-algorithms-and-homology","title":"Message-Passing Algorithms and Homology","arxiv_id":"2009.11631","date":"2020-09-24","proceeding":null,"authors":["Olivier Peltre"],"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","url_abs":"http://arxiv.org/abs/2009.11631v1","url_pdf":"http://arxiv.org/pdf/2009.11631v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"message-passing-algorithms-and-homology","repo_url":"https://github.com/opeltre/topos","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}