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

Scarlett, Jonathan; Cevher, Volkan

doi:10.1109/icassp.2017.7953326

Cited by 15 publications

(15 citation statements)

References 24 publications

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“…We begin with the case that only false positives are allowed. This setting is closely related to that of list decoding, which was studied in [9], [31], [42], [43]. Recall that asymptotic notation such as →, o(·), O(·) is with respect to p → ∞, and we assume that k → ∞ with k = o(p).…”

Section: A General Partial Recovery Achievability Resultsmentioning

confidence: 99%

Near-Optimal Noisy Group Testing via Separate Decoding of Items

Scarlett

Cevher²

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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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.

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Section: A General Partial Recovery Achievability Resultsmentioning

confidence: 99%

Near-Optimal Noisy Group Testing via Separate Decoding of Items

Scarlett

Cevher²

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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“…We briefly mention that extensions of Theorem 5.1 are given in [164] to a list decoding setting, in which the decoder outputs a list of length L ≥ k and it is only required that the list contains (1 − γ)k defectives. (The concept of list decoding for group testing also appeared much earlier under the zero-error criterion, e.g., see [110]).…”

Section: Partial Recoverymentioning

confidence: 99%

“…If L is much larger than k, this means that we are potentially allowing a large number of false positives. However, a finding of [164] is that this relaxation often only amounts to a replacement of k log 2 n k by k log 2 n L in the required number of tests (asymptotically), which is a rather minimal gain.…”

Section: Partial Recoverymentioning

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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“…We need to show that there exists some β > 1 such that f (1) (1). Hence, to guarantee that there exists some β > 1 such that βf (β) − f (1) ≤ 1, it suffices to show that (β − 1)f (1) ≤ 1 for some β > 1.…”

Section: Proof Of Lemmamentioning

confidence: 99%

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

Karimi

Kazemi

Heidarzadeh

et al. 2019

2019 IEEE Information Theory Workshop (ITW)

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This paper considers the problem of Quantitative Group Testing (QGT). Consider a set of N items among which K items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. In this work, we propose a nonadaptive QGT algorithm using sparse graph codes over biregular bipartite graphs with left-degree ℓ and right degree r and binary t-error-correcting BCH codes. The proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, we show that for the sub-linear regime where K N vanishes as K, N → ∞, the proposed algorithm requires at most m = c(t)K t log 2 ℓN c(t)K + 1 + 1 + 1 tests to recover all the defective items with probability approaching one as K, N → ∞, where c(t) depends only on t. The results of our theoretical analysis reveal that the minimum number of required tests is achieved by t = 2. The encoding and decoding of the proposed algorithm for any t ≤ 4 have the computational complexity of O(K log 2 N K ) and O(K log N K ), respectively. Our simulation results also show that the proposed algorithm significantly outperforms a non-adaptive semi-quantitative group testing algorithm recently proposed by Abdalla et al. in terms of the required number of tests for identifying all the defective items with high probability.

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

Cited by 15 publications

References 24 publications

Near-Optimal Noisy Group Testing via Separate Decoding of Items

Near-Optimal Noisy Group Testing via Separate Decoding of Items

Group Testing: An Information Theory Perspective

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

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