Amin Coja‐Oghlan

Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth et al.

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.

Amin Coja-Oghlan, Remco van der Hofstad, Lena Krieg et al.

We study the Threshold Group Testing (TGT) problem without a gap in the noiseless, non-adaptive setting, where the goal is to exactly recover a sparse binary vector from pooled test outcomes using as few tests as possible. In TGT, a test applied to a subset of items returns a positive outcome if the number of defective items in the subset reaches a prescribed threshold, and a negative outcome otherwise. Under the assumption of an analytic condition, TGT has been shown to undergo a sharp information-theoretic phase transition for exact recovery on the class of constant-column test designs. In this paper, we develop an efficient inference algorithm that achieves exact recovery with high probability using the minimum number of non-adaptive tests that are needed for the constant-column design, thereby matching the information-theoretic threshold of a natural benchmark test design. Our approach is based on a spatially coupled test design and admits a significantly simpler analysis than existing algorithms for related group testing problems. In particular, unlike previous methods for binary group testing, our algorithm does not rely on the analysis of intricate weighted sums. This leads to a more straightforward proof technique, while still allowing near-optimal performance guarantees.