Note on Noisy Group Testing: Asymptotic Bounds and Belief Propagation Reconstruction

TL;DR

This paper extends information theoretic bounds for group testing to noisy scenarios with false positives and negatives, and proposes a belief propagation algorithm for practical detection.

Contribution

It introduces bounds for noisy group testing with both false positives and negatives and develops a belief propagation method for defect detection.

Findings

Improved bounds for false negatives case

Belief propagation performs close to theoretical limits

Extension of noiseless results to noisy settings

Abstract

An information theoretic perspective on group testing problems has recently been proposed by Atia and Saligrama, in order to characterise the optimal number of tests. Their results hold in the noiseless case, where only false positives occur, and where only false negatives occur. We extend their results to a model containing both false positives and false negatives, developing simple information theoretic bounds on the number of tests required. Based on these bounds, we obtain an improved order of convergence in the case of false negatives only. Since these results are based on (computationally infeasible) joint typicality decoding, we propose a belief propagation algorithm for the detection of defective items and compare its actual performance to the theoretical bounds.