# Solving Statistical Mechanics on Sparse Graphs with Feedback Set Variational Autoregressive Networks

## TL;DR

This paper introduces a neural network-based method that simplifies sparse graph problems in statistical mechanics by extracting a feedback vertex set, enabling accurate free energy estimation and sampling.

## Contribution

It presents a novel approach combining feedback vertex set extraction with variational autoregressive networks for solving sparse graph statistical mechanics problems.

## Key findings

- Outperforms existing methods on sparse spin glasses.
- More accurate than belief-propagation on random graphs and real-world networks.
- Faster and more accurate than recent variational autoregressive networks on 2D lattices.

## Abstract

We propose a method for solving statistical mechanics problems defined on sparse graphs. It extracts a small Feedback Vertex Set (FVS) from the sparse graph, converting the sparse system to a much smaller system with many-body and dense interactions with an effective energy on every configuration of the FVS, then learns a variational distribution parameterized using neural networks to approximate the original Boltzmann distribution. The method is able to estimate free energy, compute observables, and generate unbiased samples via direct sampling without auto-correlation. Extensive experiments show that our approach is more accurate than existing approaches for sparse spin glasses. On random graphs and real-world networks, our approach significantly outperforms the standard methods for sparse systems such as the belief-propagation algorithm; on structured sparse systems such as two-dimensional lattices our approach is significantly faster and more accurate than recently proposed variational autoregressive networks using convolution neural networks.