Graphical Fermat's Principle and Triangle-Free Graph Estimation

TL;DR

This paper introduces a novel graphical Fermat's principle for estimating high-dimensional, triangle-free graphs, enabling efficient recovery of the true graph structure using a greedy algorithm based on a distribution-dependent pseudo-metric.

Contribution

The paper proposes a new regularization principle for triangle-free graph estimation and demonstrates a computationally efficient greedy algorithm for accurate graph recovery.

Findings

Greedy strategy successfully recovers true graphs

Algorithm outperforms minimum spanning tree in efficiency

Method applicable to discrete and nonparametric distributions

Abstract

We consider the problem of estimating undirected triangle-free graphs of high dimensional distributions. Triangle-free graphs form a rich graph family which allows arbitrary loopy structures but 3-cliques. For inferential tractability, we propose a graphical Fermat's principle to regularize the distribution family. Such principle enforces the existence of a distribution-dependent pseudo-metric such that any two nodes have a smaller distance than that of two other nodes who have a geodesic path include these two nodes. Guided by this principle, we show that a greedy strategy is able to recover the true graph. The resulting algorithm only requires a pairwise distance matrix as input and is computationally even more efficient than calculating the minimum spanning tree. We consider graph estimation problems under different settings, including discrete and nonparametric distribution families. Thorough numerical results are provided to illustrate the usefulness of the proposed method.