Estimating Ising Models in Total Variation Distance

2025-11-25

AI Summary

This paper studies parameter estimation for $n$-variable Ising models under total variation (TV) distance, given $l$ independent samples. We consider two broad model classes—those whose interaction matrices satisfy either bounded operator norm or $ell_infty$ norm constraints—and develop the first unified pseudo-maximum-likelihood estimation (PMLE) framework for TV-distance-based estimation. Methodologically, our analysis integrates modified logarithmic Sobolev inequalities, tensorization techniques, measure decomposition, and convergence analysis of Glauber dynamics, thereby ensuring both statistical optimality and polynomial-time computability. Under diverse structural assumptions—including tree-structured graphs, Gaussian couplings, and regimes with small eigenvalues—the proposed estimator achieves optimal or near-optimal sample complexity. To the best of our knowledge, this work provides the first theoretical and algorithmic guarantees for TV-distance-based Ising model estimation that are simultaneously general, statistically efficient, and computationally tractable.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Security and Privacy: Large-scale security measurements

Abstract

We consider the problem of estimating Ising models over $n$ variables in Total Variation (TV) distance, given $l$ independent samples from the model. While the statistical complexity of the problem is well-understood [DMR20], identifying computationally and statistically efficient algorithms has been challenging. In particular, remarkable progress has occurred in several settings, such as when the underlying graph is a tree [DP21, BGPV21], when the entries of the interaction matrix follow a Gaussian distribution [GM24, CK24], or when the bulk of its eigenvalues lie in a small interval [AJK+24, KLV24], but no unified framework for polynomial-time estimation in TV exists so far. Our main contribution is a unified analysis of the Maximum Pseudo-Likelihood Estimator (MPLE) for two general classes of Ising models. The first class includes models that have bounded operator norm and satisfy the Modified Log-Sobolev Inequality (MLSI), a functional inequality that was introduced to study the convergence of the associated Glauber dynamics to stationarity. In the second class of models, the interaction matrix has bounded infinity norm (or bounded width), which is the most common assumption in the literature for structure learning of Ising models. We show how our general results for these classes yield polynomial-time algorithms and optimal or near-optimal sample complexity guarantees in a variety of settings. Our proofs employ a variety of tools from tensorization inequalities to measure decompositions and concentration bounds.

Problem

• Estimating Ising models in total variation distance from samples
• Developing computationally efficient algorithms for general Ising models
• Providing unified framework for polynomial-time estimation guarantees

Innovation

• Maximum Pseudo-Likelihood Estimator unified analysis
• Bounded operator norm with Modified Log-Sobolev Inequality
• Bounded infinity norm interaction matrix framework