Phase Transitions in Group Testing

Scarlett, Jonathan; Cevher, Volkan

doi:10.1137/1.9781611974331.ch4

Cited by 83 publications

(165 citation statements)

References 23 publications

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“…We mention that an alternative approach was taken in [163], bearing a stronger resemblance to the above achievability proof and again relying on change-of-measure techniques from the channel coding literature. The proof of [163] has the advantage of recovering the so-called 'strong converse' (see Remark 1.3), but it requires additional effort in ensuring that the suitable sums of information densities concentrate around the corresponding conditional mutual information.…”

Section: Discussion Of Converse Proofmentioning

confidence: 99%

“…We state the main achievability result of Scarlett and Cevher [163] as follows, and then discuss the proof. where h(ρ) is the binary entropy function in bits, and c(ρ, α) is a continuous function with c(ρ, 0) > 1 − h(ρ).…”

Section: Achievability In General Sparse Regimesmentioning

confidence: 99%

“…An explicit expression for c(ρ, α) is given in [163], but it is omitted here since it is complicated and does not provide additional insight. Most interesting is the fact that Theorem 4.6 provides the exact capacity (even for non-Bernoulli and possibly adaptive tests) for all α ∈ (0, α 0 ), where α 0 is strictly positive but may depend on ρ.…”

Section: Achievability In General Sparse Regimesmentioning

confidence: 99%

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Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

Self Cite

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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: Discussion Of Converse Proofmentioning

confidence: 99%

Section: Achievability In General Sparse Regimesmentioning

confidence: 99%

Section: Achievability In General Sparse Regimesmentioning

confidence: 99%

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Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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“…The lower bound of Ω((1 − ǫ)d log(n/d)/(1 − H(σ)) for ǫ-error group testing, [15] is met (up to constant factors) by [11,16,17].…”

Section: Related Workmentioning

confidence: 99%

“…While there is significant literature on multiple alternative models of group testing [3][4][5][6][7][8][9][10][11], the focus of this work is primarily on non-adaptive group testing, under ǫ-error and zero-error reconstruction guarantee metrics. We thus restrict the discussion of prior work to the literature on lower bounds and algorithms (both deterministic and randomized) for ǫ-error and zero-error non-adaptive group testing.…”

Section: Related Workmentioning

confidence: 99%

Nearly optimal sparse group testing

Gandikota

Grigorescu

Jaggi

et al. 2016

2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Group testing is the process of pooling arbitrary subsets from a set of n items so as to identify, with a minimal number of disjunctive tests, a "small" subset of d defective items. In "classical" non-adaptive group testing, it is known that when d = o(n 1−δ ) for any δ > 0, θ(d log(n)) tests are both information-theoretically necessary, and sufficient to guarantee recovery with high probability. Group testing schemes in the literature meeting this bound require most items to be tested Ω(log(n)) times, and most tests to incorporate Ω(n/d) items.Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be "sparse". Specifically, we consider (separately) scenarios in which (a) items are finitely divisible and hence may participate in at most γ tests; and (b) tests are size-constrained to pool no more than ρ items per test. For both scenarios we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In particular, one of our main results shows that γ-finite divisibility of items forces any group testing algorithm with probability of recovery error at most ǫ to perform at least Ω(γd(n/d) (1−2ǫ)/((1+2ǫ)γ) ) tests. Analogously, for ρ-sized constrained tests, we show an information-theoretic lower bound of Ω(n log(n/d)/(ρ log(n/ρd))). In both scenarios we provide both randomized constructions (under both ǫ-error and zero-error reconstruction guarantees) and explicit constructions of computationally efficient group-testing algorithms (under ǫ-error reconstruction guarantees) that require a number of tests that are optimal up to constant factors in some regimes of n, d, γ and ρ. We also investigate the effect of unreliability/noise in test outcomes.

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Almost separable matrices

Aldridge

Baldassini²,

Gunderson

2015

J Comb Optim

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An m × n matrix A with column supports {Si} is k-separable if the disjunctions i∈K Si are all distinct over all sets K of cardinality k. While a simple counting bound shows that m > k log 2 n/k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker definition of 'almost k-separability', which requires that the disjunctions are 'mostly distinct'. We show using a random construction that these matrices exist with m = O(k log n) rows, which is optimal for k = O(n 1−β ). Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing.

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Phase Transitions in Group Testing

Cited by 83 publications

References 23 publications

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective

Nearly optimal sparse group testing

Almost separable matrices

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