Learning Loosely Connected Markov Random Fields

TL;DR

This paper introduces a new algorithm for learning the structure of loosely connected Markov random fields, capable of handling short cycles and requiring logarithmic sample complexity, with applications to various graphical models.

Contribution

The paper presents a novel conditional independence test based algorithm that effectively learns the structure of loosely connected Markov random fields, including general Ising models, with improved computational efficiency.

Findings

Algorithm correctly identifies edges in graphs with short cycles.

Sample complexity is logarithmic in the number of nodes.

Achieves comparable or better complexity than previous methods.

Abstract

We consider the structure learning problem for graphical models that we call loosely connected Markov random fields, in which the number of short paths between any pair of nodes is small, and present a new conditional independence test based algorithm for learning the underlying graph structure. The novel maximization step in our algorithm ensures that the true edges are detected correctly even when there are short cycles in the graph. The number of samples required by our algorithm is C*log p, where p is the size of the graph and the constant C depends on the parameters of the model. We show that several previously studied models are examples of loosely connected Markov random fields, and our algorithm achieves the same or lower computational complexity than the previously designed algorithms for individual cases. We also get new results for more general graphical models, in particular, our algorithm learns general Ising models on the Erdos-Renyi random graph G(p, c/p) correctly with running time O(np^5).