Estimation of High Dimensional Bounded Discrete Graphical Models via Regularized Generalized Score Matching

2026-06-25

AI Summary

This work addresses the challenges posed by the intractable normalization constant in high-dimensional, unbounded discrete graphical models, which complicates parameter constraints and conditional dependence modeling. To circumvent this issue, the authors propose a bounded discrete graphical model that inherently avoids normalization difficulties and introduce an unnormalized estimation method based on Regularized Generalized Score Matching (BRIDGE). By reparameterizing variables to restore curvature in the loss function, the approach overcomes objective degeneracy. Under non-convex settings, the method establishes a population-level separability property that substitutes for global convexity, enabling exact support recovery in high dimensions. Theoretical analysis provides non-asymptotic bounds on estimation error and guarantees graph structure consistency. Experimental results demonstrate that BRIDGE is stable, computationally efficient, and yields highly interpretable outcomes.

Abstract

Graphical models for multivariate count data are widely used to characterize conditional dependence structures. For count variables with unbounded support, however, ensuring a finite normalizing constant typically imposes restrictive constraints on interaction parameters. We propose bounded discrete graphical models for multivariate discrete responses with finite support, which remove such constraints by construction while retaining interpretable dependence on the observed scale. We develop a regularized generalized score matching estimator (BRIDGE), which provides a normalization-free surrogate for likelihood-based estimation. The approach yields a unified system of estimating equations for all parameters and enables joint regularization through an $\ell_1$ penalty. To address degeneracy in the loss geometry, we introduce a reparameterization that restores curvature along the intercept direction and facilitates stable computation. On the theoretical side, we analyze a nonconvex objective and establish a population separation property that replaces global convexity. This yields nonasymptotic estimation error bounds and exact support recovery in high-dimensional regimes. Simulation studies and real data analyses demonstrate that BRIDGE accurately recovers graph structure and provides a stable and interpretable framework for high-dimensional discrete graphical modeling.

Problem

• discrete graphical models
• high-dimensional estimation
• conditional dependence
• finite support
• normalizing constant

Innovation

• bounded discrete graphical models
• regularized generalized score matching
• normalization-free estimation
• reparameterization
• high-dimensional support recovery