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Lecture

Graphical models: Inference and Factor Graphs

In course

PHYS-642: Statistical physics for optimization & learning

This course covers the statistical physics approach to computer science problems, with an emphasis on heuristic & rigorous mathematical technics, ranging from graph theory and constraint satisfaction to inference to machine learning, neural networks and st ...

Description

This lecture covers the topic of graphical models, focusing on inference and factor graphs. It explains the concept of graphical models, interactions, and factor graphs, illustrating with examples and equations. The lecture discusses the use of factor graphs for modeling complex systems and making probabilistic inferences.

Instructors (2)

Florent Gérard Krzakala

Lenka Zdeborová

Lenka Zdeborová is a Professor of Physics and of Computer Science in École Polytechnique Fédérale de Lausanne where she leads the Statistical Physics of Computation Laboratory. She received a PhD in physics from University Paris-Sud and from Charles University in Prague in 2008. She spent two years in the Los Alamos National Laboratory as the Director's Postdoctoral Fellow. Between 2010 and 2020 she was a researcher at CNRS working in the Institute of Theoretical Physics in CEA Saclay, France. In 2014, she was awarded the CNRS bronze medal, in 2016 Philippe Meyer prize in theoretical physics and an ERC Starting Grant, in 2018 the Irène Joliot-Curie prize, in 2021 the Gibbs lectureship of AMS. She is an editorial board member for Journal of Physics A, Physical Review E, Physical Review X, SIMODS, Machine Learning: Science and Technology, and Information and Inference. Lenka's expertise is in applications of concepts from statistical physics, such as advanced mean field methods, replica method and related message-passing algorithms, to problems in machine learning, signal processing, inference and optimization. She enjoys erasing the boundaries between theoretical physics, mathematics and computer science.

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (32)

Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets and may be thought of as a coloring of the graph with two colors: if one colors all nodes in blue, and all nodes in red, each edge has endpoints of differing colors, as is required in the graph coloring problem.

Factor graph

A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to represent factorization of a probability distribution function, enabling efficient computations, such as the computation of marginal distributions through the sum-product algorithm. One of the important success stories of factor graphs and the sum-product algorithm is the decoding of capacity-approaching error-correcting codes, such as LDPC and turbo codes.

Factor-critical graph

In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph) is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.) A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.

Graph factorization

In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization partitions the edges of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is a proper edge coloring with k colors. A 2-factor is a collection of cycles that spans all vertices of the graph.

Petersen graph

In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by .

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