# Generalized Precision Matrix for Scalable Estimation of Nonparametric Markov Networks

arXiv: https://arxiv.org/abs/2305.11379 · HTML: https://mlsanity.com/paper/2305.11379/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 52 references that are in the map.
- First posted on arXiv 19 May 2023.

## Builds on (most cited first, 11 of 11)

- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [On Graphical Models via Univariate Exponential Family Distributions](https://mlsanity.com/paper/1301.4183/) (2013)
- [Support recovery without incoherence: A case for nonconvex regularization](https://mlsanity.com/paper/1412.5632/) (2014)
- [Estimation of High-Dimensional Graphical Models Using Regularized Score Matching](https://mlsanity.com/paper/1507.00433/) (2015)
- [Reliable Causal Discovery with Improved Exact Search and Weaker Assumptions](https://mlsanity.com/paper/2201.05666/) (2022)
- [Learning non-Gaussian graphical models via Hessian scores and triangular transport](https://mlsanity.com/paper/2101.03093/) (2021)
- [Beyond normality: Learning sparse probabilistic graphical models in the non-Gaussian setting](https://mlsanity.com/paper/1711.00950/) (2017)
- [The Statistical Physics of Real-World Networks](https://mlsanity.com/paper/1810.05095/) (2018)
- [High-dimensional Mixed Graphical Models](https://mlsanity.com/paper/1304.2810/) (2013)

## Built on by (most cited first, 7 of 7)

- [Causal Representation Learning from Multiple Distributions: A General Setting](https://mlsanity.com/paper/2402.05052/) (2024)
- [Learning General Causal Structures with Hidden Dynamic Process for Climate Analysis](https://mlsanity.com/paper/2501.12500/) (2025)
- [Nonparametric Identification of Latent Concepts](https://mlsanity.com/paper/2510.00136/) (2025)
- [Higher-Order Causal Structure Learning with Additive Models](https://mlsanity.com/paper/2511.03831/) (2025)
- [A General Representation-Based Approach to Multi-Source Domain Adaptation](https://mlsanity.com/paper/2604.23790/) (2026)
- [Understanding Breast Cancer Survival: Using Causality and Language Models on Multi-omics Data](https://mlsanity.com/paper/2305.18410/) (2023)
- [Covariate-dependent Graphical Model Estimation via Neural Networks with Statistical Guarantees](https://mlsanity.com/paper/2504.16356/) (2025)

## Abstract (arXiv)

A Markov network characterizes the conditional independence structure, or Markov property, among a set of random variables. Existing work focuses on specific families of distributions (e.g., exponential families) and/or certain structures of graphs, and most of them can only handle variables of a single data type (continuous or discrete). In this work, we characterize the conditional independence structure in general distributions for all data types (i.e., continuous, discrete, and mixed-type) with a Generalized Precision Matrix (GPM). Besides, we also allow general functional relations among variables, thus giving rise to a Markov network structure learning algorithm in one of the most general settings. To deal with the computational challenge of the problem, especially for large graphs, we unify all cases under the same umbrella of a regularized score matching framework. We validate the theoretical results and demonstrate the scalability empirically in various settings.