A New Family of Tractable Ising Models

TL;DR

This paper introduces a new family of zero-field Ising models constructed by combining planar and small components, enabling polynomial-time inference and sampling, and extends tractability to certain non-planar models.

Contribution

It proposes a novel construction of Ising models that are tractable, with efficient algorithms for inference and sampling, extending beyond planar cases.

Findings

Developed an $O(N^{3/2})$ algorithm for K5-minor-free models.

Demonstrated improved approximation for non-zero field Ising models.

Extended tractability to models beyond genus- and treewidth-bounded cases.

Abstract

We present a new family of zero-field Ising models over N binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components along with subsets of at most three vertices into a tree. The polynomial time algorithm of the dynamic programming type for solving exact inference (partition function computation) and sampling consists of a sequential application of an efficient (for planar) or brute-force (for $O(1)$-sized) inference and sampling to the components as a black box. To illustrate the utility of the new family of tractable graphical models, we first build an $O(N^{3/2})$ algorithm for inference and sampling of the K5-minor-free zero-field Ising models - an extension of the planar zero-field Ising models - which is neither genus- nor treewidth-bounded. Second, we demonstrate empirically an improvement in the approximation quality of the NP-hard problem of the square-grid Ising model (with non-zero field) inference.