On Graphical Models and Convex Geometry

TL;DR

This paper presents a novel method called `betaMix' for detecting significant correlations in high-dimensional graphical models using convex geometry, without assuming network sparsity or specific distributional structures.

Contribution

It introduces a mixture-model based approach leveraging convex geometry theorems to control error rates in edge detection for diverse data distributions.

Findings

Effective control of error rates in edge detection

Applicable to both light-tailed and heavy-tailed distributions

No assumptions on network sparsity or structure

Abstract

We introduce a mixture-model of beta distributions to identify significant correlations among $P$ predictors when $P$ is large. The method relies on theorems in convex geometry, which we use to show how to control the error rate of edge detection in graphical models. Our `betaMix' method does not require any assumptions about the network structure, nor does it assume that the network is sparse. The results in this article hold for a wide class of data generating distributions that include light-tailed and heavy-tailed spherically symmetric distributions.