Exact Thresholds for Noisy Non-Adaptive Group Testing

TL;DR

This paper determines the precise asymptotic thresholds for noisy non-adaptive group testing under common random test designs, advancing understanding beyond previous partial results.

Contribution

It derives exact thresholds for noisy group testing with two test designs, using novel analysis of maximum-likelihood decoding and impossibility results.

Findings

Established exact asymptotic thresholds for noisy group testing

Analyzed maximum-likelihood decoding to derive upper bounds

Proved impossibility results for certain failure events

Abstract

In recent years, the mathematical limits and algorithmic bounds for probabilistic group testing have become increasingly well-understood, with exact asymptotic thresholds now being known in general scaling regimes for the noiseless setting. In the noisy setting where each test outcome is flipped with constant probability, there have been similar developments, but the overall understanding has lagged significantly behind the noiseless setting. In this paper, we substantially narrow this gap by deriving exact asymptotic thresholds for the noisy setting under two widely-studied random test designs: i.i.d. Bernoulli and near-constant tests-per-item. These thresholds are established by combining components of an existing information-theoretic threshold decoder with a novel analysis of maximum-likelihood decoding (upper bounds), and deriving a novel set of impossibility results by analyzing certain failure events for optimal maximum-likelihood decoding (lower bounds).