Learning local neighborhoods of non-Gaussian graphical models: A measure transport approach

TL;DR

This work addresses learning conditional independence structure in high-dimensional, potentially non-Gaussian data by introducing L-SING, a local approach that learns per-variable transport maps $S^k$ to model $X_k|X_{-k}$ and constructs a generalized precision matrix $oldsymbol{AOmega}$ to infer edges. By parameterizing $S^k$ with Unconstrained Monotonic Neural Networks and optimizing a KL-based objective, L-SING generalizes neighborhood selection in Gaussian settings and encompasses nonparanormal models, enabling scalable, parallelizable graph recovery. The method is validated on Gaussian, non-Gaussian (but structured) benchmarks and a high-dimensional gene-expression dataset (156 genes, 578 samples), demonstrating accurate structure recovery, superior handling of nonlinear dependencies, and practical scalability. The results suggest L-SING offers a principled, memory-efficient alternative to global density estimation for non-Gaussian graphical models, with potential extensions to mixed-type data and improved edge-thresholding strategies.

Abstract

Identifying the Markov properties or conditional independencies of a collection of random variables is a fundamental task in statistics for modeling and inference. Existing approaches often learn the structure of a probabilistic graphical model, which encodes these dependencies, by assuming that the variables follow a distribution with a simple parametric form. Moreover, the computational cost of many algorithms scales poorly for high-dimensional distributions, as they need to estimate all the edges in the graph simultaneously. In this work, we propose a scalable algorithm to infer the conditional independence relationships of each variable by exploiting the local Markov property. The proposed method, named Localized Sparsity Identification for Non-Gaussian Distributions (L-SING), estimates the graph by using flexible classes of transport maps to represent the conditional distribution for each variable. We show that L-SING includes existing approaches, such as neighborhood selection with Lasso, as a special case. We demonstrate the effectiveness of our algorithm in both Gaussian and non-Gaussian settings by comparing it to existing methods. Lastly, we show the scalability of the proposed approach by applying it to high-dimensional non-Gaussian examples, including a biological dataset with more than 150 variables.

Figures (9)

• Figure 1: (a) The undirected graphical model; (b) Adjacency matrix of true graph (white corresponds to an edge, black to no edge) for the 10-dimensional Gaussian distribution.
• Figure 2: Generalized precision matrix for L-SING and the estimated precision amtrix for GLASSO computed using $N = 10,000$ Gaussian evaluation samples.
• Figure 3: Sensitivity of false positives with L-SING for different thresholds and sample sizes on the Gaussian data.
• Figure 4: (a) The undirected graphical model; (b) Adjacency matrix of true graph for the butterfly distribution.
• Figure 5: Estimated generalized precision matrix for the Butterfly distribution with $d=10$ and $d=40$ variables using L-SING with $N = 10,000$ evaluation samples.

Theorems & Definitions (6)

• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Theorem A.1
• proof