Clique Matrices for Statistical Graph Decomposition and Parameterising Restricted Positive Definite Matrices

TL;DR

This paper introduces clique matrices as a versatile tool for graph decomposition and positive definite matrix parameterization, enabling structured analysis and approximation of complex graph-based matrices.

Contribution

It presents clique matrices as a new representation for graphs, facilitating statistical decomposition and structured parameterization of positive definite matrices.

Findings

Clique matrices effectively decompose graphs into well-connected clusters.

They can parameterize all positive definite matrices with zero constraints.

Clique matrices enable structured Factor Analysis approximations.

Abstract

We introduce Clique Matrices as an alternative representation of undirected graphs, being a generalisation of the incidence matrix representation. Here we use clique matrices to decompose a graph into a set of possibly overlapping clusters, de ned as well-connected subsets of vertices. The decomposition is based on a statistical description which encourages clusters to be well connected and few in number. Inference is carried out using a variational approximation. Clique matrices also play a natural role in parameterising positive de nite matrices under zero constraints on elements of the matrix. We show that clique matrices can parameterise all positive de nite matrices restricted according to a decomposable graph and form a structured Factor Analysis approximation in the non-decomposable case.