Clique Matrices for Statistical Graph Decomposition and Parameterising Restricted Positive Definite Matrices
This work addresses graph decomposition and matrix parameterization problems in statistics and machine learning, presenting a novel method that could be useful for researchers in these fields, though it appears incremental as it builds on existing representations like incidence matrices.
The authors tackled the problem of decomposing undirected graphs into overlapping clusters by introducing Clique Matrices as a generalization of incidence matrices, using a statistical description to encourage well-connected and few clusters, with inference via variational approximation, and applied this to parameterize positive definite matrices under zero constraints for decomposable graphs and as a structured Factor Analysis approximation for non-decomposable cases.
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.