Poisson Group Testing: A Probabilistic Model for Boolean Compressed Sensing

TL;DR

This paper introduces Poisson group testing, a probabilistic framework for Boolean compressed sensing where the number of defectives follows a Poisson distribution, providing bounds and algorithms for efficient identification.

Contribution

It develops a new Poisson-based model for group testing, deriving bounds and proposing test designs and algorithms that approach theoretical limits.

Findings

Lower bounds on the number of tests for nonadaptive methods

Test matrix constructions close to the lower bounds

Stage-wise algorithm nearly optimal in expected tests

Abstract

We introduce a novel probabilistic group testing framework, termed Poisson group testing, in which the number of defectives follows a right-truncated Poisson distribution. The Poisson model has a number of new applications, including dynamic testing with diminishing relative rates of defectives. We consider both nonadaptive and semi-adaptive identification methods. For nonadaptive methods, we derive a lower bound on the number of tests required to identify the defectives with a probability of error that asymptotically converges to zero; in addition, we propose test matrix constructions for which the number of tests closely matches the lower bound. For semi-adaptive methods, we describe a lower bound on the expected number of tests required to identify the defectives with zero error probability. In addition, we propose a stage-wise reconstruction algorithm for which the expected number of tests is only a constant factor away from the lower bound. The methods rely only on an estimate of the average number of defectives, rather than on the individual probabilities of subjects being defective.