# Deep Gaussian Markov Random Fields

arXiv: https://arxiv.org/abs/2002.07467 · HTML: https://mlsanity.com/paper/2002.07467/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2020.

## Built on by (most cited first, 8 of 8)

- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [A Sparse Expansion For Deep Gaussian Processes](https://mlsanity.com/paper/2112.05888/) (2021)
- [Scalable Deep Gaussian Markov Random Fields for General Graphs](https://mlsanity.com/paper/2206.05032/) (2022)
- [Efficient methods for Gaussian Markov random fields under sparse linear constraints](https://mlsanity.com/paper/2106.01712/) (2021)
- [Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transport](https://mlsanity.com/paper/2201.09485/) (2022)
- [Deep Gaussian Markov Random Fields for Graph-Structured Dynamical Systems](https://mlsanity.com/paper/2306.08445/) (2023)
- [Synchronous Multi-view Neural Diffusion](https://mlsanity.com/paper/2609.39019/) (2026)

## Abstract (arXiv)

Gaussian Markov random fields (GMRFs) are probabilistic graphical models widely used in spatial statistics and related fields to model dependencies over spatial structures. We establish a formal connection between GMRFs and convolutional neural networks (CNNs). Common GMRFs are special cases of a generative model where the inverse mapping from data to latent variables is given by a 1-layer linear CNN. This connection allows us to generalize GMRFs to multi-layer CNN architectures, effectively increasing the order of the corresponding GMRF in a way which has favorable computational scaling. We describe how well-established tools, such as autodiff and variational inference, can be used for simple and efficient inference and learning of the deep GMRF. We demonstrate the flexibility of the proposed model and show that it outperforms the state-of-the-art on a dataset of satellite temperatures, in terms of prediction and predictive uncertainty.