Algorithms for Convex Hull Finding in Undirected Graphical Models

Abstract

An undirected graphical model is a joint distribution family defined on an undirected graph, and the convex hull of a node set in the graph is the minimal convex subgraph containing it. It has been shown that a graphical model is collapsible onto the minimal local sub-model induced by the convex hull which contains variables of interest under Gaussian and multinomial distributions. This motivates many scholars to design algorithms for finding the unique convex hull containing nodes of interest in a graph. In this paper, we propose two algorithms called, respectively, the node absorption algorithm (NA) and the inducing path absorption algorithm (IPA), to find the minimal convex subgraph containing variables of interest in an undirected graph. These algorithms can be used as potential tools to find the minimal sub-model including variables of interest onto which a graphical model of large-scale can be collapsible. Experiments show that the proposed IPA significantly outperforms the NA and other existing algorithms. Furthermore, we apply the IPA to a gene network so as to collapse a large network onto a smaller network including the interested variables, and thus to achieve the aim of structural dimension reduction.