Model-Based and Graph-Based Priors for Group Testing

TL;DR

This paper explores how incorporating structural priors, including subset-based and Ising model-based dependencies, can reduce the number of tests needed for effective group testing, introducing new decoding methods and theoretical insights.

Contribution

It introduces a novel Ising model-based prior for group testing and develops corresponding decoding algorithms, advancing the understanding of prior-informed testing strategies.

Findings

Reduced number of tests with structural priors

Integer Quadratic Program decoder performs well empirically

Theoretical limits are affected by prior assumptions

Abstract

The goal of the group testing problem is to identify a set of defective items within a larger set of items, using suitably-designed tests whose outcomes indicate whether any defective item is present. In this paper, we study how the number of tests can be significantly decreased by leveraging the structural dependencies between the items, i.e., by incorporating prior information. To do so, we pursue two different perspectives: (i) As a generalization of the uniform combinatorial prior, we consider the case that the defective set is uniform over a \emph{subset} of all possible sets of a given size, and study how this impacts the information-theoretic limits on the number of tests for approximate recovery; (ii) As a generalization of the i.i.d.~prior, we introduce a new class of priors based on the Ising model, where the associated graph represents interactions between items. We show that this naturally leads to an Integer Quadratic Program decoder, which can be converted to an Integer Linear Program and/or relaxed to a non-integer variant for improved computational complexity, while maintaining strong empirical recovery performance.