Noise-Resilient Group Testing: Limitations and Constructions

TL;DR

This paper explores the limits and constructions of noise-resilient group testing schemes for sparse Boolean vectors, demonstrating how approximate reconstruction can surpass traditional bounds even with high measurement unreliability.

Contribution

It introduces new randomized and explicit measurement schemes that enable approximate reconstruction of sparse vectors with high noise tolerance, breaking previous bounds for exact recovery.

Findings

Approximate reconstruction allows surpassing the information-theoretic lower bound for exact recovery.

Proposed schemes achieve $O(d \,\log n)$ measurements with high noise resilience.

Explicit constructions enable fast, near-linear time reconstruction.

Abstract

We study combinatorial group testing schemes for learning $d$-sparse Boolean vectors using highly unreliable disjunctive measurements. We consider an adversarial noise model that only limits the number of false observations, and show that any noise-resilient scheme in this model can only approximately reconstruct the sparse vector. On the positive side, we take this barrier to our advantage and show that approximate reconstruction (within a satisfactory degree of approximation) allows us to break the information theoretic lower bound of $\tilde{\Omega}(d^2 \log n)$ that is known for exact reconstruction of $d$-sparse vectors of length $n$ via non-adaptive measurements, by a multiplicative factor $\tilde{\Omega}(d)$. Specifically, we give simple randomized constructions of non-adaptive measurement schemes, with $m=O(d \log n)$ measurements, that allow efficient reconstruction of $d$-sparse vectors up to $O(d)$ false positives even in the presence of $\delta m$ false positives and $O(m/d)$ false negatives within the measurement outcomes, for any constant $\delta < 1$. We show that, information theoretically, none of these parameters can be substantially improved without dramatically affecting the others. Furthermore, we obtain several explicit constructions, in particular one matching the randomized trade-off but using $m = O(d^{1+o(1)} \log n)$ measurements. We also obtain explicit constructions that allow fast reconstruction in time $\poly(m)$, which would be sublinear in $n$ for sufficiently sparse vectors. The main tool used in our construction is the list-decoding view of randomness condensers and extractors.