Learning Loosely Connected Markov Random Fields
This addresses the problem of efficiently learning graphical model structures for researchers in machine learning and statistics, offering incremental improvements over existing algorithms.
The paper tackles the structure learning problem for loosely connected Markov random fields by introducing a new conditional independence test-based algorithm that correctly identifies true edges even with short cycles, requiring only C*log p samples where p is the graph size. It achieves the same or lower computational complexity than prior methods for specific models and extends to more general cases, such as learning Ising models on Erdos-Renyi graphs with O(np^5) running time.
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).