# Learning Networks from Gaussian Graphical Models and Gaussian Free Fields

arXiv: https://arxiv.org/abs/2308.02344 · HTML: https://mlsanity.com/paper/2308.02344/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 61 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 7 of 7)

- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Learning Some Popular Gaussian Graphical Models without Condition Number Bounds](https://mlsanity.com/paper/1905.01282/) (2019)
- [Exact recovery in the Ising blockmodel](https://mlsanity.com/paper/1612.03880/) (2016)
- [Sparse covariance matrix estimation in high-dimensional deconvolution](https://mlsanity.com/paper/1710.10870/) (2017)
- [Learning with latent group sparsity via heat flow dynamics on networks](https://mlsanity.com/paper/2201.08326/) (2022)

## Abstract (arXiv)

We investigate the problem of estimating the structure of a weighted network from repeated measurements of a Gaussian Graphical Model (GGM) on the network. In this vein, we consider GGMs whose covariance structures align with the geometry of the weighted network on which they are based. Such GGMs have been of longstanding interest in statistical physics, and are referred to as the Gaussian Free Field (GFF). In recent years, they have attracted considerable interest in the machine learning and theoretical computer science. In this work, we propose a novel estimator for the weighted network (equivalently, its Laplacian) from repeated measurements of a GFF on the network, based on the Fourier analytic properties of the Gaussian distribution. In this pursuit, our approach exploits complex-valued statistics constructed from observed data, that are of interest on their own right. We demonstrate the effectiveness of our estimator with concrete recovery guarantees and bounds on the required sample complexity. In particular, we show that the proposed statistic achieves the parametric rate of estimation for fixed network size. In the setting of networks growing with sample size, our results show that for Erdos-Renyi random graphs G(d,p) above the connectivity threshold, we demonstrate that network recovery takes place with high probability as soon as the sample size n satisfies n gg d4 log d · p-2.