Combinatorial Inference for Graphical Models

TL;DR

This paper introduces a new framework for combinatorial inference in graphical models, focusing on testing global graph properties rather than parameter estimation, with theoretical limits and practical algorithms demonstrated on synthetic and real data.

Contribution

It develops a unified theory for the fundamental limits of combinatorial graph inference and proposes practical algorithms that match these theoretical bounds.

Findings

Established minimax lower bounds for combinatorial graph tests.

Designed algorithms that achieve these bounds in practice.

Validated methods on synthetic and brain network data.

Abstract

We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity, the presence of a cycle of certain size, or the maximum degree of the graph. To begin with, we develop a unified theory for the fundamental limits of a large family of combinatorial inference problems. We propose new concepts including structural packing and buffer entropies to characterize how the complexity of combinatorial graph structures impacts the corresponding minimax lower bounds. On the other hand, we propose a family of novel and practical structural testing algorithms to match the lower bounds. We provide thorough numerical results on both synthetic graphical models and brain networks to illustrate the usefulness of these proposed methods.