Computationally sufficient statistics for Ising models

TL;DR

This paper explores the limits of learning Ising models from limited statistics, showing that model parameters can be efficiently reconstructed using only low-order sufficient statistics, especially with prior structural information.

Contribution

It introduces a method to learn Ising model parameters using limited sufficient statistics proportional to the model's width, improving efficiency over traditional full-sample methods.

Findings

Parameters can be reconstructed from statistics up to order O(γ).

Model structure and couplings can be inferred with limited observations.

Prior structural information further reduces observational requirements.

Abstract

Learning Gibbs distributions using only sufficient statistics has long been recognized as a computationally hard problem. On the other hand, computationally efficient algorithms for learning Gibbs distributions rely on access to full sample configurations generated from the model. For many systems of interest that arise in physical contexts, expecting a full sample to be observed is not practical, and hence it is important to look for computationally efficient methods that solve the learning problem with access to only a limited set of statistics. We examine the trade-offs between the power of computation and observation within this scenario, employing the Ising model as a paradigmatic example. We demonstrate that it is feasible to reconstruct the model parameters for a model with $\ell_1$ width $\gamma$ by observing statistics up to an order of $O(\gamma)$. This approach allows us to infer the model's structure and also learn its couplings and magnetic fields. We also discuss a setting where prior information about structure of the model is available and show that the learning problem can be solved efficiently with even more limited observational power.