Noisy Non-Adaptive Group Testing: A (Near-)Definite Defectives Approach

TL;DR

This paper introduces Near-Definite Defectives algorithms for noisy group testing, providing bounds on the number of tests needed and analyzing different noise models, including Z-channel and symmetric noise.

Contribution

It proposes a new class of algorithms for noisy group testing and derives bounds and converse results, extending analysis to general binary noise models.

Findings

Bounds on the number of tests for vanishing error probability.

Matching achievable rates and converse bounds under certain noise channels.

Improvements over existing bounds in broad regimes for binary noise models.

Abstract

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly-noisy tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. We study the noisy version of this problem, where the outcome of each standard noiseless group test is subject to independent noise, corresponding to passing the noiseless result through a binary channel. We introduce a class of algorithms that we refer to as Near-Definite Defectives (NDD), and study bounds on the required number of tests for asymptotically vanishing error probability under Bernoulli random test designs. In addition, we study algorithm-independent converse results, giving lower bounds on the required number of tests under Bernoulli test designs. Under reverse Z-channel noise, the achievable rates and converse results match in a broad range of sparsity regimes, and under Z-channel noise, the two match in a narrower range of dense/low-noise regimes. We observe that although these two channels have the same Shannon capacity when viewed as a communication channel, they can behave quite differently when it comes to group testing. Finally, we extend our analysis of these noise models to a general binary noise model (including symmetric noise), and show improvements over known existing bounds in broad scaling regimes.