# Learning non-Gaussian graphical models via Hessian scores and triangular transport

arXiv: https://arxiv.org/abs/2101.03093 · HTML: https://mlsanity.com/paper/2101.03093/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 86 references that are in the map.
- First posted on arXiv 8 Jan 2021.

## Builds on (most cited first, 16 of 16)

- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [On Nesting Monte Carlo Estimators](https://mlsanity.com/paper/1709.06181/) (2017)
- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [On Graphical Models via Univariate Exponential Family Distributions](https://mlsanity.com/paper/1301.4183/) (2013)
- [Estimation of High-Dimensional Graphical Models Using Regularized Score Matching](https://mlsanity.com/paper/1507.00433/) (2015)
- [Beyond normality: Learning sparse probabilistic graphical models in the non-Gaussian setting](https://mlsanity.com/paper/1711.00950/) (2017)
- [The Gaussian Graphical Model in Cross-sectional and Time-series Data](https://mlsanity.com/paper/1609.04156/) (2016)
- [A layered multiple importance sampling scheme for focused optimal Bayesian experimental design](https://mlsanity.com/paper/1903.11187/) (2019)
- [An Improved Modified Cholesky Decomposition Method for Precision Matrix Estimation](https://mlsanity.com/paper/1710.05163/) (2017)

## Built on by (most cited first, 16 of 16)

- [Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Scheme](https://mlsanity.com/paper/2211.16757/) (2022)
- [Ensemble transport smoothing. Part II: Nonlinear updates](https://mlsanity.com/paper/2210.17435/) (2022)
- [Generalized Precision Matrix for Scalable Estimation of Nonparametric Markov Networks](https://mlsanity.com/paper/2305.11379/) (2023)
- [Ensemble transport smoothing. Part I: Unified framework](https://mlsanity.com/paper/2210.17000/) (2022)
- [Localized Diffusion Models](https://mlsanity.com/paper/2505.04417/) (2025)
- [Diagonal Nonlinear Transformations Preserve Structure in Covariance and Precision Matrices](https://mlsanity.com/paper/2107.04136/) (2021)
- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Causal Effect Identification in Uncertain Causal Networks](https://mlsanity.com/paper/2208.04627/) (2022)
- [Learning local neighborhoods of non-Gaussian graphical models: A measure transport approach](https://mlsanity.com/paper/2503.13899/) (2025)
- [An adaptive ensemble filter for heavy-tailed distributions: tuning-free inflation and localization](https://mlsanity.com/paper/2310.08741/) (2023)
- [Causal Order Discovery based on Monotonic SCMs](https://mlsanity.com/paper/2410.19870/) (2024)
- [Covariate-dependent Graphical Model Estimation via Neural Networks with Statistical Guarantees](https://mlsanity.com/paper/2504.16356/) (2025)
- [Nonparametric learning of heterogeneous graphical model on network-linked data](https://mlsanity.com/paper/2507.01473/) (2025)
- [Conditional Independence Estimates for the Generalized Nonparanormal](https://mlsanity.com/paper/2508.11050/) (2025)
- [Graphical conditional generative modeling for digital twin modeling](https://mlsanity.com/paper/2606.16219/) (2026)
- [Learning the Structure of Triangular Transport Maps](https://mlsanity.com/paper/2609.37122/) (2026)

## Abstract (arXiv)

Undirected probabilistic graphical models represent the conditional dependencies, or Markov properties, of a collection of random variables. Knowing the sparsity of such a graphical model is valuable for modeling multivariate distributions and for efficiently performing inference. While the problem of learning graph structure from data has been studied extensively for certain parametric families of distributions, most existing methods fail to consistently recover the graph structure for non-Gaussian data. Here we propose an algorithm for learning the Markov structure of continuous and non-Gaussian distributions. To characterize conditional independence, we introduce a score based on integrated Hessian information from the joint log-density, and we prove that this score upper bounds the conditional mutual information for a general class of distributions. To compute the score, our algorithm SING estimates the density using a deterministic coupling, induced by a triangular transport map, and iteratively exploits sparse structure in the map to reveal sparsity in the graph. For certain non-Gaussian datasets, we show that our algorithm recovers the graph structure even with a biased approximation to the density. Among other examples, we apply SING to learn the dependencies between the states of a chaotic dynamical system with local interactions.