Non-Adaptive Group Testing based on Sparse Pooling Graphs

TL;DR

This paper provides an information theoretic analysis of non-adaptive group testing using sparse pooling graphs, establishing conditions for accurate object status estimation in noiseless and noisy scenarios.

Contribution

It introduces direct part theorems for the existence of estimators with arbitrarily small error, based on averaging bounds over regular pooling graph ensembles.

Findings

Sharp threshold behaviors observed in asymptotic regimes

Conditions for estimator existence established for noiseless and noisy cases

Numerical results support theoretical predictions

Abstract

In this paper, an information theoretic analysis on non-adaptive group testing schemes based on sparse pooling graphs is presented. The binary status of the objects to be tested are modeled by i.i.d. Bernoulli random variables with probability p. An (l, r, n)-regular pooling graph is a bipartite graph with left node degree l and right node degree r, where n is the number of left nodes. Two scenarios are considered: a noiseless setting and a noisy one. The main contributions of this paper are direct part theorems that give conditions for the existence of an estimator achieving arbitrary small estimation error probability. The direct part theorems are proved by averaging an upper bound on estimation error probability of the typical set estimator over an (l,r, n)-regular pooling graph ensemble. Numerical results indicate sharp threshold behaviors in the asymptotic regime.