The capacity of adaptive group testing

Baldassini, Leonardo; Johnson, Oliver; Aldridge, Matthew

doi:10.1109/isit.2013.6620712

Cited by 91 publications

(146 citation statements)

References 20 publications

(46 reference statements)

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“…This definition was first proposed for the combinatorial case by Baldassini, Aldridge and Johnson [20], and extended to the general case (see Definition 5.2) in [120]. This definition generalizes a similar earlier definition of rate by Malyutov [143,144], which applied only in the very sparse (k constant) regime.…”

Section: Counting Bound and Ratementioning

confidence: 91%

Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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The group testing problem concerns discovering a small number of defective items within a large population by performing tests on pools of items. A test is positive if the pool contains at least one defective, and negative if it contains no defectives. This is a sparse inference problem with a combinatorial flavour, with applications in medical testing, biology, telecommunications, information technology, data science, and more.In this monograph, we survey recent developments in the group testing problem from an information-theoretic perspective. We cover several related developments: efficient algorithms with practical storage and computation requirements, achievability bounds for optimal decoding methods, and algorithmindependent converse bounds. We assess the theoretical guarantees not only in terms of scaling laws, but also in terms of the constant factors, leading to the notion of the rate of group testing, indicating the amount of information learned per test. Considering both noiseless and noisy settings, we identify several regimes where existing algorithms are provably optimal or near-optimal, as well as regimes where there remains greater potential for improvement.In addition, we survey results concerning a number of variations on the standard group testing problem, including partial recovery criteria, adaptive algorithms with a limited number of stages, constrained test designs, and sublineartime algorithms.S 0 , S 1 partition of the defective set (Equation (4.4)) ı information density (Equation (4.6)) X 0,τ , X 1,τ sub-matrices of X corresponding to (S 0 , S 1 ) with |S 0 | = τ (Equation (4.14))τ ) (Equation (4.16)) 2. If the preceding test is negative, A contains no defective items, and we halt. If the test is positive, continue. 3. If A consists of a single item, then that item is defective, and we halt. Otherwise, pick half of the items in A, and call this set B. Perform a single test of the pool B. 4. If the test is positive, set A := B. If the test is negative, set A := A \ B. Return to Step 3. A BRIEF REVIEW OF ZERO-ERROR GROUP TESTINGObserve that S(i) is the set of tests containing item i, while S(K) is the set of positive tests when the defective set is K.Definition 1.10. A matrix X is called k-separable if the support unions S(L) are distinct over all subsets L ⊆ {1, 2, . . . , n} of size |L| = k.A matrix X is calledk-separable if the support unions S(L) are distinct over all subsets L ⊆ {1, 2, . . . , n} of size |L| ≤ k.Clearly, using a k-separable matrix as a test design ensures that group testing will provide different outcomes for each possible defective set of size k; thus,

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Section: Counting Bound and Ratementioning

confidence: 91%

Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

Self Cite

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“…Early studies of this type were performed by Malyutov [9], and more recent studies include those of Atia and Saligrama [12], Aldridge et al [11,14], and Laarhoven [15]. In the case that the number of defective items k does not scale with p, the fundamental limits are well-understood for both the noiseless and noisy settings [9,12,15]; for example, in the noiseless case, the smallest possible number of measurements with vanishing error probability behaves as k log 2 p k (1 + o(1)), which is in fact the same threshold as that for optimal adaptive measurements [16] (i.e., designs for which each test may depend on previous outcomes). Surprisingly, there remain significant gaps in the best known upper and lower bounds on n when k scales with p, which is of considerable interest in applications where the number of defective items is "not too small".…”

Section: )mentioning

confidence: 99%

“…The converse bound shown has been proved (using significantly different techniques) even for optimal adaptive measurements [16], and hence a key implication is that adaptivity provides no asymptotic gain over non-adaptive Bernoulli measurements when…”

Section: Problem Statementmentioning

confidence: 99%

Phase Transitions in Group Testing

Scarlett¹,

Cevher²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

159

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The group testing problem consists of determining a sparse subset of a set of items that are "defective" based on a set of possibly noisy tests, and arises in areas such as medical testing, fault detection, communication protocols, pattern matching, and database systems. We study the fundamental limits of any group testing procedure regardless of its computational complexity. In the noiseless case with the number of defective items k scaling with the total number of items p as O(p θ ) (θ ∈ (0, 1)), we show that the probability of reconstruction error tends to one when n ≤ k log 2 (1)), for some explicit constant c(θ). For θ ≤ 1 3 , we show that c(θ) = 1, thus providing an exact threshold on the required number measurements, i.e. a phase transition, which was previously known only in the limit as θ → 0. Analogous necessary and sufficient conditions are derived for the noisy setting, and also for a relaxed partial recovery criterion.

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“…in the non-adaptive setting, while in the adaptive setting the same is true with k = O(p θ ) for any θ ∈ (0, 1) [16]. Various partial recovery and list decoding results have also appeared previously.…”

Section: Previous Workmentioning

confidence: 79%

How little does non-exact recovery help in group testing?

Scarlett

Cevher

2017

2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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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 tests. We characterize the informationtheoretic performance limits in the presence of list decoding, in which the decoder may output a list containing more elements than the number of defectives, and the only requirement is that the true defective set is a subset of the list, or more generally, that their overlap exceeds a given threshold. We show that even under this highly relaxed criterion, in several scaling regimes the asymptotic number of tests is no smaller than the exact recovery setting. However, we also provide examples where a reduction is provably attained. We support our theoretical findings with numerical experiments.

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The capacity of adaptive group testing

Cited by 91 publications

References 20 publications

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective

Phase Transitions in Group Testing

How little does non-exact recovery help in group testing?

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