Converse Bounds for Noisy Group Testing with Arbitrary Measurement Matrices

TL;DR

This paper establishes new theoretical limits for noisy group testing with arbitrary measurement matrices, providing bounds that match known achievable results and extend understanding beyond i.i.d. measurement scenarios.

Contribution

It introduces two converse bounds applicable to arbitrary measurement matrices and general noise models, advancing the theoretical understanding of group testing limits.

Findings

Strong converse bound matches existing achievability in several cases.

Weak converse bound extends matching bounds to more general scenarios.

Results apply to arbitrary measurement matrices and noise models.

Abstract

We consider the group testing problem, in which one seeks to identify a subset of defective items within a larger set of items based on a number of noisy tests. While matching achievability and converse bounds are known in several cases of interest for i.i.d.~measurement matrices, less is known regarding converse bounds for arbitrary measurement matrices. We address this by presenting two converse bounds for arbitrary matrices and general noise models. First, we provide a strong converse bound ($\mathbb{P}[\mathrm{error}] \to 1$) that matches existing achievability bounds in several cases of interest. Second, we provide a weak converse bound ($\mathbb{P}[\mathrm{error}] \not\to 0$) that matches existing achievability bounds in greater generality.