Group Testing with Random Pools: Phase Transitions and Optimal Strategy

Mézard, Marc; Tarzia, Marco; Toninelli, Cristina

doi:10.1007/s10955-008-9528-9

Cited by 31 publications

(39 citation statements)

References 23 publications

(60 reference statements)

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“…However, some of the arguments therein are based on a "no short loops" assumption that is only verified rigorously for θ ≥ . In contrast, in this paper we obtain phase transitions for θ ≤ 1 3 , which does not overlap with the range of interest in [20]. In fact, it is verified numerically in [20] that the assumption regarding short loops is invalid for θ = 1 3 .…”

Section: )contrasting

confidence: 63%

“…In contrast, in this paper we obtain phase transitions for θ ≤ 1 3 , which does not overlap with the range of interest in [20]. In fact, it is verified numerically in [20] that the assumption regarding short loops is invalid for θ = 1 3 . Finally, we briefly comment on practical decoders.…”

Section: )contrasting

confidence: 63%

“…Another particularly related work is that of Mézard et al [20], who derived phase transitions for various random measurement designs in the noiseless setting, for certain scaling regimes. However, some of the arguments therein are based on a "no short loops" assumption that is only verified rigorously for θ ≥ .…”

Section: )mentioning

confidence: 99%

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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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Section: )contrasting

confidence: 63%

Section: )contrasting

confidence: 63%

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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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“…Again, L = Θ(T /k) (or equivalently, m = Θ(n/k)) is a useful scaling. We will not focus on these designs in this monograph, but mention that they were studied in the papers [148,192], among others.…”

Section: Basic Definitions and Notationmentioning

confidence: 99%

“…While Definition 2.3 suffices for our purposes, it is worth mentioning that it is one of a variety of related randomized designs that have appeared in the literature. Indeed, a variety of works have considered (exactly-)constant testsper-item (see for example [136,148]). There is evidence that such matrices provide similar gains, but to our knowledge, this has not been proved in the same generality and rigour as the case of near-constant tests-per-item.…”

Section: Improved Rates With Near-constant Tests-peritemmentioning

confidence: 99%

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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Random and quasi-random designs in group testing

Noonan

Zhigljavsky

2022

Journal of Statistical Planning and Inference

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Group Testing with Random Pools: Phase Transitions and Optimal Strategy

Cited by 31 publications

References 23 publications

Phase Transitions in Group Testing

Phase Transitions in Group Testing

Group Testing: An Information Theory Perspective

Random and quasi-random designs in group testing

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