Statistical and Computational Phase Transitions in Group Testing

TL;DR

This paper investigates phase transitions in group testing, establishing sharp thresholds for detection and recovery, and explores the computational limits of polynomial-time algorithms, revealing a computational-statistical gap.

Contribution

It provides new information-theoretic bounds, precise test thresholds for different designs, and analyzes the computational limits of efficient algorithms in group testing.

Findings

Sharp all-or-nothing phase transition in the constant-column design.

Precise bounds on the number of tests needed for detection in Bernoulli design.

Evidence of a computational-statistical gap at low sparsity levels.

Abstract

We study the group testing problem where the goal is to identify a set of k infected individuals carrying a rare disease within a population of size n, based on the outcomes of pooled tests which return positive whenever there is at least one infected individual in the tested group. We consider two different simple random procedures for assigning individuals to tests: the constant-column design and Bernoulli design. Our first set of results concerns the fundamental statistical limits. For the constant-column design, we give a new information-theoretic lower bound which implies that the proportion of correctly identifiable infected individuals undergoes a sharp "all-or-nothing" phase transition when the number of tests crosses a particular threshold. For the Bernoulli design, we determine the precise number of tests required to solve the associated detection problem (where the goal is to distinguish between a group testing instance and pure noise), improving both the upper and lower bounds of Truong, Aldridge, and Scarlett (2020). For both group testing models, we also study the power of computationally efficient (polynomial-time) inference procedures. We determine the precise number of tests required for the class of low-degree polynomial algorithms to solve the detection problem. This provides evidence for an inherent computational-statistical gap in both the detection and recovery problems at small sparsity levels. Notably, our evidence is contrary to that of Iliopoulos and Zadik (2021), who predicted the absence of a computational-statistical gap in the Bernoulli design.