Near-Optimal Noisy Group Testing via Separate Decoding of Items

Abstract

The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. In this paper, we revisit an efficient algorithm for noisy group testing in which each item is decoded separately (Malyutov and Mateev, 1980), and develop novel performance guarantees via an information-theoretic framework for general noise models. For the special cases of no noise and symmetric noise, we find that the asymptotic number of tests required for vanishing error probability is within a factor log 2 ≈ 0.7 of the informationtheoretic optimum at low sparsity levels, and that with a small fraction of allowed incorrectly decoded items, this guarantee extends to all sublinear sparsity levels. In addition, we provide a converse bound showing that if one tries to move slightly beyond our low-sparsity achievability threshold using separate decoding of items and i.i.d. randomized testing, the average number of items decoded incorrectly approaches that of a trivial decoder.

Noisy Adaptive Group Testing: Bounds and Algorithms

“…For the achievability part, we propose an adaptive algorithm whose first stage can be taken as any non-adaptive algorithm that comes with partial recovery guarantees, and whose second stage (and third stage in a refined version) improve this initial estimate. By letting the first stage use the information-theoretic threshold decoder of [8], we attain an achievability bound that is near-tight in many cases of interest, whereas by using separate decoding of items as per [13], [16], we attain a slightly weaker guarantee while still maintaining computational efficiency. In addition, we provide a novel converse bound showing that Ω(k log k) tests are always necessary, and hence, the implied constant in any scaling of the form n = Θ k log p k with k = Θ(p θ ) must grow unbounded as θ → 1.…”

“…• The method of separate decoding of items, also known as separate testing of inputs [13], [16], also considers the items separately, but uses all of the tests. Specifically, a given item's status is selected via a binary hypothesis test.…”

“…Specifically, a given item's status is selected via a binary hypothesis test. This method was studied for k = O(1) in [13], and for k = Θ(p θ ) in [16]; in particular, it was shown that the number of tests is within a factor log 2 of the optimal information-theoretic threshold under exact recovery as θ → 0, and under partial recovery (with d max = Θ(k)) for all θ ∈ (0, 1).…”

Noisy Non-Adaptive Group Testing: A (Near-)Definite Defectives Approach

“…• In the non-adaptive setting with symmetric noise, it was shown in [10], [22] that an information-theoretic threshold decoder attains the bound (14) when k = Θ(p θ ) for sufficiently small θ > 0. The analysis of [15,Appendix A] shows that analogous findings also hold for the Z and RZ noise models.…”

“…Specifically, a given item's status is selected via a binary hypothesis test. This method was studied for k = O(1) in [20], and for k = Θ(p θ ) in [14]. In particular, it was shown that for the symmetric noise model, the number of tests is within a factor log 2 of the optimal information-theoretic threshold as θ → 0.…”