Scalable and non-iterative graphical model estimation

TL;DR

This paper introduces a fast, non-iterative method for estimating positive definite graphical models in high dimensions, overcoming the scalability and robustness limitations of traditional iterative methods like IPF.

Contribution

The paper presents a novel non-iterative approach for high-dimensional graphical model estimation with proven efficiency, statistical guarantees, and robustness, addressing key limitations of existing methods.

Findings

Outperforms state-of-the-art methods in computational complexity as dimension increases.

Compatible with scalable sparsity selection techniques.

Achieves high statistical precision and robustness in numerical experiments.

Abstract

Graphical models have found widespread applications in many areas of modern statistics and machine learning. Iterative Proportional Fitting (IPF) and its variants have become the default method for undirected graphical model estimation, and are thus ubiquitous in the field. As the IPF is an iterative approach, it is not always readily scalable to modern high-dimensional data regimes. In this paper we propose a novel and fast non-iterative method for positive definite graphical model estimation in high dimensions, one that directly addresses the shortcomings of IPF and its variants. In addition, the proposed method has a number of other attractive properties. First, we show formally that as the dimension p grows, the proportion of graphs for which the proposed method will outperform the state-of-the-art in terms of computational complexity and performance tends to 1, affirming its efficacy in modern settings. Second, the proposed approach can be readily combined with scalable non-iterative thresholding-based methods for high-dimensional sparsity selection. Third, the proposed method has high-dimensional statistical guarantees. Moreover, our numerical experiments also show that the proposed method achieves scalability without compromising on statistical precision. Fourth, unlike the IPF, which depends on the Gaussian likelihood, the proposed method is much more robust.