# Mutual information of spin systems from autoregressive neural networks

arXiv: https://arxiv.org/abs/2304.13412 · HTML: https://mlsanity.com/paper/2304.13412/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 58 references that are in the map.
- First posted on arXiv 26 Apr 2023.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Gauge-equivariant flow models for sampling in lattice field theories with pseudofermions](https://mlsanity.com/paper/2207.08945/) (2022)
- [Flow-based sampling in the lattice Schwinger model at criticality](https://mlsanity.com/paper/2202.11712/) (2022)
- [Hierarchical autoregressive neural networks for statistical systems](https://mlsanity.com/paper/2203.10989/) (2022)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)
- [Simulating first-order phase transition with hierarchical autoregressive networks](https://mlsanity.com/paper/2212.04955/) (2022)
- [Gradient estimators for normalising flows](https://mlsanity.com/paper/2202.01314/) (2022)

## Built on by (most cited first, 4 of 4)

- [Hierarchical autoregressive neural networks in three-dimensional statistical system](https://mlsanity.com/paper/2503.08610/) (2025)
- [Training normalizing flows with computationally intensive target probability distributions](https://mlsanity.com/paper/2308.13294/) (2023)
- [Sampling two-dimensional spin systems with transformers](https://mlsanity.com/paper/2604.27738/) (2026)
- [Variational Autoregressive Networks with probability priors](https://mlsanity.com/paper/2605.16020/) (2026)

## Abstract (arXiv)

We describe a new direct method to estimate bipartite mutual information of a classical spin system based on Monte Carlo sampling enhanced by autoregressive neural networks. It allows studying arbitrary geometries of subsystems and can be generalized to classical field theories. We demonstrate it on the Ising model for four partitionings, including a multiply-connected even-odd division. We show that the area law is satisfied for temperatures away from the critical temperature: the constant term is universal, whereas the proportionality coefficient is different for the even-odd partitioning.