Learning Some Popular Gaussian Graphical Models without Condition Number Bounds

TL;DR

This paper introduces polynomial-time algorithms for learning certain Gaussian Graphical Models (GGMs) without requiring well-conditioned precision matrices, enabling structure recovery even with strong variable dependencies.

Contribution

The paper presents the first algorithms for attractive and walk-summable GGMs that work with minimal sample sizes and no condition number assumptions.

Findings

Algorithms successfully recover graph structures with minimal samples.

Existing methods fail with long dependency chains, our algorithms succeed.

Experimental results demonstrate robustness of the proposed methods.

Abstract

Gaussian Graphical Models (GGMs) have wide-ranging applications in machine learning and the natural and social sciences. In most of the settings in which they are applied, the number of observed samples is much smaller than the dimension and they are assumed to be sparse. While there are a variety of algorithms (e.g. Graphical Lasso, CLIME) that provably recover the graph structure with a logarithmic number of samples, they assume various conditions that require the precision matrix to be in some sense well-conditioned. Here we give the first polynomial-time algorithms for learning attractive GGMs and walk-summable GGMs with a logarithmic number of samples without any such assumptions. In particular, our algorithms can tolerate strong dependencies among the variables. Our result for structure recovery in walk-summable GGMs is derived from a more general result for efficient sparse linear regression in walk-summable models without any norm dependencies. We complement our results with experiments showing that many existing algorithms fail even in some simple settings where there are long dependency chains, whereas ours do not.