Lecture

Graphical models: Inference and Factor Graphs

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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)

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.

Florent Gérard Krzakala

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (32)

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 .

Cubic graph

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.

Biregular graph

In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in is and the degree of the vertices in is , then the graph is said to be -biregular. Every complete bipartite graph is -biregular. The rhombic dodecahedron is another example; it is (3,4)-biregular. An -biregular graph must satisfy the equation .

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