Group Testing: An Information Theory Perspective

Abstract

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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“…However, the number of tests in each search scales as log 2 n ∼ log 2 (1/p) for the optimal value of n, at low p. In contrast, the most efficient known parallel searches, called "noiseless, nonadaptive" tests in Ref. [18], require a factor of e ln 2 more tests (see their equations (2.8) and (2.10)), just as our algorithm does.…”

“…The advantage of our approach over Dorfman's is that of going to higher dimensions, and using group testing uniformly. Whereas testing procedures may be represented as matrices in Dorfman's and other approaches [18], in our approach higher dimensional tensors are required.…”

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“…It should be acknowledged that this valuable boost does not explore the theoretical and practically achievable rates of compressive sampling capabilities. While theoretically perfect, reconstruction of original test results available with not much more than k log 2 (N/k) measurements ( 25 ), where N is the number of samples and k is the number of positive cases, with the use of modern decoding algorithms, the achievable rates are close to the theoretical bounds. This means a 10- to 20-fold rate increase for a 1–0.1% prevalence band is possible with more sophisticated pooling schemes and decoding algorithms.…”

“…As collecting blood samples and performing a single Wassermann test for each man appeared to be quite resource demanding in the circumstances of World War II, pooling blood samples and performing group tests was observed to be quite effective since the disease was relatively rare. Later on, group testing has become a popular topic in the information theory field, enabling orders of magnitude saving from the test numbers while being able to pinpoint sparse positives accurately ( 25 ). Similarly, the attractiveness of recovering sparse signals from a small number of measurements led, in the mid-2000s, to the birth of an entire research area called compressive sampling (compressed sensing) in the signal processing field around ( 26 ).…”

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“…Because the positive samples are indistinguishable from negative samples, a test must be performed on a sample or a group of samples in order to determine their status. The test is typically assumed to always be accurate, even when many samples are tested together (in practice, this is often not the case and approaches that consider test error and constraints on the number of samples per pool have been examined [ 19 , 20 ]). In the worst case, all of the samples would need to be tested individually requiring N tests.…”