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

Graphical Fermat's Principle and Triangle-Free Graph Estimation