Inference and Sampling of $K_{33}$-free Ising Models
This work addresses computational challenges in statistical physics and machine learning for Ising models, offering incremental improvements by generalizing beyond planar graphs.
The authors tackled the problem of efficiently computing partition functions and sampling configurations for Ising models, extending tractability from planar graphs to $K_{33}$-free topologies, resulting in polynomial-time algorithms for these more complex structures.
We call an Ising model tractable when it is possible to compute its partition function value (statistical inference) in polynomial time. The tractability also implies an ability to sample configurations of this model in polynomial time. The notion of tractability extends the basic case of planar zero-field Ising models. Our starting point is to describe algorithms for the basic case computing partition function and sampling efficiently. To derive the algorithms, we use an equivalent linear transition to perfect matching counting and sampling on an expanded dual graph. Then, we extend our tractable inference and sampling algorithms to models, whose triconnected components are either planar or graphs of $O(1)$ size. In particular, it results in a polynomial-time inference and sampling algorithms for $K_{33}$ (minor) free topologies of zero-field Ising models - a generalization of planar graphs with a potentially unbounded genus.