Semiquantitative Group Testing

Emad, Amin; Milenković, Olgica

doi:10.1109/tit.2014.2327630

Cited by 35 publications

(34 citation statements)

References 53 publications

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“…• The semi-quantitative model [68] lies in between the two extremes of the standard model and the linear model. The model has a number of thresholds, and we get to observe the largest of those thresholds that the number of defective items in the test |{i ∈ K : x i = 1}| is greater than or equal to.…”

Section: Other Group Testing Modelsmentioning

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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Section: Other Group Testing Modelsmentioning

confidence: 99%

Group Testing: An Information Theory Perspective

Aldridge¹,

Johnson²,

Scarlett³

2019

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“…In [18] and [19], we introduced SQ-separable test matrices to identify the defectives with zero error-probability. A [q; Q; η; (l ∶ u); e]-SQ-separable code is a q-ary matrix for a SQGT model with thresholds η = [0, η 1 , η 2 , .…”

Section: Superimposed Codes For Sqgt and Summary Of The Resultsmentioning

confidence: 99%

“…However, due to the limited precision in obtaining the measurements, the linearity assumption of the measurements does not apply in many practical applications. In [27], [28], [29], quantized compressed sensing (QCS) was introduced to overcome the limitation of infinite precision in CS, while in [18], [19], we introduced the semi-quantitative group testing (SQGT) paradigm to overcome this issue in the the GT framework. The motivation for using quantized linear measurements in GT stems from applications in genotyping [19] and conflict resolution in multiple access channel (MAC) communication.…”

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confidence: 99%

“…In [27], [28], [29], quantized compressed sensing (QCS) was introduced to overcome the limitation of infinite precision in CS, while in [18], [19], we introduced the semi-quantitative group testing (SQGT) paradigm to overcome this issue in the the GT framework. The motivation for using quantized linear measurements in GT stems from applications in genotyping [19] and conflict resolution in multiple access channel (MAC) communication. Since a detailed description of these applications and the motivation behind SQGT is provided in [19], we refer the interested reader to this publication for more details.…”

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confidence: 99%

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Code Construction and Decoding Algorithms for Semi-Quantitative Group Testing With Nonuniform Thresholds

Emad

Milenković

2016

IEEE Trans. Inform. Theory

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We analyze a new group testing scheme, termed semi-quantitative group testing, which may be viewed as a concatenation of an adder channel and a discrete quantizer. Our focus is on non-uniform quantizers with arbitrary thresholds. For the most general semi-quantitative group testing model, we define three new families of sequences capturing the constraints on the code design imposed by the choice of the thresholds. The sequences represent extensions and generalizations of B h and certain types of super-increasing and lexicographically ordered sequences, and they lead to code structures amenable for efficient recursive decoding. We describe the decoding methods and provide an accompanying computational complexity and performance analysis.

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“…Many other models have been proposed in the literature, such as the adder channel, also known as quantitative GT [8], threshold GT [11], and symmetric GT [12]. More recent developments include the semi-quantitative group testing (SQGT) paradigm, which provides a unifying framework for a number of GT models and generalizes the notion of GT to nonbinary test matrices and non-binary test outcomes [13], [14]. In addition, GT is closely related to compressed sensing (CS) [15], [16] and integer CS [17]; the main differences lie in the structure of the alphabet used (R or C in CS, {0, 1} or a discrete set of integers for GT, and a bounded set of integers for integer CS) and the operations used to perform dimensionality reduction (addition in CS and integer CS, Boolean OR in conventional GT).…”

Section: Introductionmentioning

confidence: 99%

Poisson Group Testing: A Probabilistic Model for Boolean Compressed Sensing

Emad

Milenković

2015

IEEE Trans. Signal Process.

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We introduce a novel probabilistic group testing framework, termed Poisson group testing, in which the number of defectives follows a right-truncated Poisson distribution. The Poisson model has a number of new applications, including dynamic testing with diminishing relative rates of defectives. We consider both nonadaptive and semi-adaptive identification methods. For nonadaptive methods, we derive a lower bound on the number of tests required to identify the defectives with a probability of error that asymptotically converges to zero; in addition, we propose test matrix constructions for which the number of tests closely matches the lower bound. For semiadaptive methods, we describe a lower bound on the expected number of tests required to identify the defectives with zero error probability. In addition, we propose a stage-wise reconstruction algorithm for which the expected number of tests is only a constant factor away from the lower bound. The methods rely only on an estimate of the average number of defectives, rather than on the individual probabilities of subjects being defective.

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Semiquantitative Group Testing

Cited by 35 publications

References 53 publications

Group Testing: An Information Theory Perspective

Group Testing: An Information Theory Perspective

Code Construction and Decoding Algorithms for Semi-Quantitative Group Testing With Nonuniform Thresholds

Poisson Group Testing: A Probabilistic Model for Boolean Compressed Sensing

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