Corpus ID: 221879066

Message-Passing Algorithms and Homology.

  title={Message-Passing Algorithms and Homology.},
  author={Olivier Peltre},
  journal={arXiv: Mathematical Physics},
  • Olivier Peltre
  • Published 24 September 2020
  • Mathematics, Physics
  • arXiv: Mathematical Physics
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… Expand
1 Citations
Belief Propagation as Diffusion
The purpose of this text is to describe the structure of belief networks as concisely as possible, with the geometric operations that appear in the rewriting of BP equations. Expand


Constructing free-energy approximations and generalized belief propagation algorithms
This work explains how to obtain region-based free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms, and describes empirical results showing that GBP can significantly outperform BP. Expand
Convergence of the cluster-variation method in the thermodynamic limit
The cluster-variation method (CVM) is discussed in the thermodynamic limit of an infinitely extended lattice. The relationship between the variational principle for the free energy per lattice pointExpand
Topological Information Data Analysis
Methods that quantify the structure of statistical interactions within a given data set, and propose that higher-order statistical interactions and non-identically distributed variables are constitutive characteristics of biological systems that should be estimated in order to unravel their significant statistical structure and diversity are presented. Expand
Loopy Belief Propagation for Approximate Inference: An Empirical Study
This paper compares the marginals computed using loopy propagation to the exact ones in four Bayesian network architectures, including two real-world networks: ALARM and QMR, and finds that the loopy beliefs often converge and when they do, they give a good approximation to the correct marginals. Expand
Consistent Families of Measures and Their Extensions
Let $\Sigma $ be a family of Borel fields of subsets of a set S and $\mu_\mathfrak{S} $ probabilistic measures on measurable spaces $\langle {\mathfrak{S},S} \rangle $, where $\mathfrak{S} \in \SigmaExpand
The graphical brain: Belief propagation and active inference
This paper formulate neuronal processing as belief propagation under deep generative models that can entertain both discrete and continuous states, leading to distinct schemes for belief updating that play out on the same (neuronal) architecture. Expand
Belief Propagation and Revision in Networks with Loops
It is shown that for all networks with a single loop, the MAP estimate obtained by belief revision at convergence is guaranteed to give the globally optimal sequence of states and the result is independent of the length of the cycle and the size of the state space. Expand
The sheaf-theoretic structure of non-locality and contextuality
It is shown that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections, and a linear algebraic approach to computing these obstructions is described, which allows a systematic treatment of arguments for non- Locality and contextuality. Expand
On Loopy Belief Propagation - Local Stability Analysis for Non-Vanishing Fields
This work obtains all fixed points of belief propagation and performs a local stability analysis, and explains the close connections between the underlying graph structure, the existence of multiple solutions, and the capability of belief propagate to converge. Expand
Bethe free energy, Kikuchi approximations, and belief propagation algorithms
This is anupdatedandexpandedversionof TR2000-26,but it is still in draft form. Beliefpropagation(BP)wasonlysupposedtowork for tree-likenetworksbutworks surprisinglywell in many applicationsinvolvingExpand