Group Testing Model Research Articles

It Was “All” for “Nothing”: Sharp Phase Transitions for Noiseless Discrete Channels

We establish a phase transition known as the “all-or-nothing” phenomenon for noiseless discrete channels. This class of models includes the Bernoulli group testing model and the planted Gaussian perceptron model. Previously, the existence of the all-or-nothing phenomenon for such models was only known in a limited range of parameters. Our work extends the results to all signals with arbitrary sublinear sparsity. Over the past several years, the all-or-nothing phenomenon has been established in various models as an outcome of two seemingly disjoint results: one positive result establishing the “all” half of all-or-nothing, and one impossibility result establishing the “nothing” half. Our main technique in the present work is to show that for noiseless discrete channels, the “all” half implies the “nothing” half, that is, a proof of “all” can be turned into a proof of “nothing.” Since the “all” half can often be proven by straightforward means—for instance, by the first-moment method—our equivalence gives a powerful and general approach towards establishing the existence of this phenomenon in other contexts.

Analysis of Theoretical Bounds in Noisy Threshold Group Testing

The objective of this study was to describe a noisy threshold group testing model where positive and negative cases could occur depending on virus concentration in coronavirus disease 2019 (COVID-19) diagnosis with output results flipped due to measurement noise. We investigated lower bounds for successful reconstruction of a small set of defective samples in the noisy threshold group testing framework. To this end, using Fano’s inequality, we derived the minimum number of tests required to find unknown signals with defective samples. Our results showed that the minimum number of tests on probability of error for reconstruction of unknown signals was a function of the defective rate and noise probability. We obtained lower bounds for performance of the noisy threshold group testing framework with respect to noise intervals. In addition, the relationship between defective rate of signals and sparsity of group matrices to design optimal noisy threshold group testing systems is presented.

Theoretical Bounds on the Number of Tests in Noisy Threshold Group Testing Frameworks

We consider a variant of group testing (GT) models called noisy threshold group testing (NTGT), in which when there is more than one defective sample in a pool, its test result is positive. We deal with a variant model of GT where, as in the diagnosis of COVID-19 infection, if the virus concentration does not reach a threshold, not only do false positives and false negatives occur, but also unexpected measurement noise can reverse a correct result over the threshold to become incorrect. We aim to determine how many tests are needed to reconstruct a small set of defective samples in this kind of NTGT problem. To this end, we find the necessary and sufficient conditions for the number of tests required in order to reconstruct all defective samples. First, Fano’s inequality was used to derive a lower bound on the number of tests needed to meet the necessary condition. Second, an upper bound was found using a MAP decoding method that leads to giving the sufficient condition for reconstructing defective samples in the NTGT problem. As a result, we show that the necessary and sufficient conditions for the successful reconstruction of defective samples in NTGT coincide with each other. In addition, we show a trade-off between the defective rate of the samples and the density of the group matrix which is then used to construct an optimal NTGT framework.

Mangat Randomized Response Group Testing Theory

This paper proposed a new Randomized Response Group Testing (RRGT) model that estimates proportion of people characterizing a sensitive variable under study. Simple random sampling with replacement, binomial probability distribution and maximum likelihood were used as randomization procedure, sampling model and estimation technique, respectively. The distributional properties (expectation and variance) of the proposed estimator, efficiency comparison of the model with some existing models, and numerical illustration of all the competing models were also explored. The study found that the developed model outperformed competing existing orthodox RRMs and earlier RRGT model in terms of efficiency, privacy protection and it is economically advantageous.

Improved Bounds for Noisy Group Testing With Constant Tests per Item

The group testing problem is concerned with identifying a small set of infected individuals in a large population. At our disposal is a testing procedure that allows us to test several individuals together. In an idealized setting, a test is positive if and only if at least one infected individual is included and negative otherwise. Significant progress was made in recent years towards understanding the information-theoretic and algorithmic properties in this noiseless setting. In this paper, we consider a noisy variant of group testing where test results are flipped with certain probability, including the realistic scenario where sensitivity and specificity can take arbitrary values. Using a test design where each individual is assigned to a fixed number of tests, we derive explicit algorithmic bounds for two commonly considered inference algorithms and thereby naturally extend the results of Scarlett & Cevher (2016) and Scarlett & Johnson (2020). We provide improved performance guarantees for the efficient algorithms in these noisy group testing models – indeed, for a large set of parameter choices the bounds provided in the paper are the strongest currently proved.

Optimized Solution for Nucleic Acid Detection Based on Group Testing Model

Under the critical situation of recurrent new crown outbreaks, large-scale full-scale nucleic acid testing has become the norm across the country. When the total number of people in a region is large and the infection rate is small, mixed testing can greatly improve testing efficiency and save costs compared with one-person testing. For the optimization of the mixed testing model, the Group Testing strategy is used to analyze the mixed testing method with the minimum number of tests, build a model and calculate the results with the help of Matlab, and finally obtain the optimal ratio of mixed testing in each region.

On the All-or-Nothing Behavior of Bernoulli Group Testing

In this article, we study the problem of non-adaptive group testing, in which one seeks to identify which items are defective given a set of suitably-designed tests whose outcomes indicate whether or not at least one defective item was included in the test. The most widespread recovery criterion seeks to exactly recover the entire defective set, and relaxed criteria such as approximate recovery and list decoding have also been considered. In contrast, we study the fundamental limits of group testing under the significantly relaxed weak recovery criterion, which only seeks to identify a small fraction (e.g., 0.01) of the defective items. Given the near-optimality of i.i.d. Bernoulli testing for exact recovery in sufficiently sparse scaling regimes, it is natural to ask whether this design additionally succeeds with much fewer tests under weak recovery. Our main negative result shows that this is not the case, and in fact, under i.i.d. Bernoulli random testing in the sufficiently sparse regime, an all-or-nothing phenomenon occurs: When the number of tests is slightly below a threshold, weak recovery is impossible, whereas when the number of tests is slightly above the same threshold, high-probability exact recovery is possible. In establishing this result, we additionally prove similar negative results under Bernoulli designs for the weak detection problem (distinguishing between the group testing model vs. completely random outcomes) and the problem of identifying a single item that is definitely defective. On the positive side, we show that all three relaxed recovery criteria can be attained using considerably fewer tests under suitably-chosen non-Bernoulli designs. Thus, our results collectively indicate that when too few tests are available, naively applying i.i.d. Bernoulli testing can lead to catastrophic failure, whereas “cutting one’s losses” and adopting a more carefully-chosen design can still succeed in attaining these less stringent criteria.

Theoretical Bounds on Performance in Threshold Group Testing Schemes

A threshold group testing (TGT) scheme with lower and upper thresholds is a general model of group testing (GT) which identifies a small set of defective samples. In this paper, we consider the TGT scheme that require the minimum number of tests. We aim to find lower and upper bounds for finding a set of defective samples in a large population. The decoding for the TGT scheme is exploited by minimization of the Hamming weight in channel coding theory and the probability of error is also defined. Then, we derive a new upper bound on the probability of error and extend a lower bound from conventional one to the TGT scheme. We show that the upper and lower bounds well match with each other at the optimal density ratio of the group matrix. In addition, we conclude that when the gaps between the two thresholds in the TGT framework increase, the group matrix with a high density should be used to achieve optimal performance.

Hypothesis Test for Bounds on the Size of Random Defective Set

The conventional model of disjunctive group testing assumes that there are several defective elements (or defectives) among a large population, and a group test yields the positive response if and only if the testing group contains at least one defective element. The basic problem is to find all defectives using a minimal possible number of group tests. However, when the number of defectives is unknown there arises an additional problem, namely: how to estimate the random number of defective elements. n this paper, we concentrate on testing the hypothesis $H_0$: the number of defectives $\le s_1$ against the alternative hypothesis $H_1$: the number of defectives $\ge s_2$. We introduce a new decoding algorithm based on the comparison of the number of tests having positive responses with an appropriate fixed threshold. For some asymptotic regimes on $s_1$ and $s_2$, the proposed algorithm is shown to be order-optimal. Additionally, our simulation results verify the advantages of the proposed algorithm such as low complexity and a small error probability compared with known algorithms.

Nearly Optimal Sparse Group Testing

Group testing is the process of pooling arbitrary subsets from a set of $ {n}$ items so as to identify, with a minimal number of tests, a “small” subset of $ {d}$ defective items. In “classical” non-adaptive group testing, it is known that when $ {d}$ is substantially smaller than $ {n}$ , $\Theta ( {d}\log ( {n}))$ tests are both information-theoretically necessary and sufficient to guarantee recovery with high probability. Group testing schemes in the literature that meet this bound require most items to be tested $ {\Omega }(\log ( {n}))$ times, and most tests to incorporate $ {\Omega }({{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 1) items are finitely divisible and hence may participate in at most $ {\gamma } \in {o}(\log ( {n}))$ tests; or 2) tests are size-constrained to pool no more than $\rho \in {o}({{n/d}})$ 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 $ {\gamma }$ -finite divisibility of items forces any non-adaptive group testing algorithm with the probability of recovery error at most $ {\epsilon }$ to perform at least $ {\gamma } {d}({ {n/d}})^{({1}-{5} {\epsilon })/ {\gamma }}$ tests. Analogously, for $ {\rho }$ -sized constrained tests, we show an information-theoretic lower bound of $ {\Omega }( {n}/ {\rho })$ tests for high-probability recovery–hence in both settings the number of tests required grows dramatically (relative to the classical setting) as a function of $ {n}$ . In both scenarios, we provide both randomized constructions and explicit constructions of designs with computationally efficient reconstruction algorithms that require a number of tests that is optimal up to constant or small polynomial factors in some regimes of ${{n, d,}} {\gamma }$ , and $ {\rho }$ . The randomized design/reconstruction algorithm in the $ {\rho }$ -sized test scenario is universal –independent of the value of $ {d}$ , as long as $ {\rho } \in {o}({\textbf {n/d}})$ . We also investigate the effect of unreliability/noise in test outcomes, and show that whereas the impact of noise in test outcomes can be obviated with a small (constant factor) penalty in the number of tests in the $ {\rho }$ -sized tests scenario, there is no group-testing procedure, regardless of the number of tests, that can combat noise in the $ {\gamma }$ -divisible scenario.

A Combinatorial Model of Two-Sided Search

We study a new model of combinatorial group testing: the item to be found (a.k.a. the target) occupies an unknown node in a graph. At each time instant, we can test (or query) a subset of the nodes and learn whether the target occupies any of such nodes. Immediately after the result of the test is available, the target can move to any node adjacent to its present location. The search finishes when we are able to locate the object with some predefined accuracy [Formula: see text] (a parameter fixed beforehand), i.e., to indicate a set of [Formula: see text] nodes that includes the location of the object. In this paper we study two types of problems related to the above model: (i) what is the minimum value of the accuracy parameter for which a search strategy in the above sense exists; (ii) given the accuracy, what is the minimum number of tests that allow to locate the target. We study these questions on paths, cycles, and trees as underlying graphs and provide tight answers for the above questions. We also consider a restricted variant of the problem, where the number of moves of the target is bounded.

Adaptive search for one defective in the additive group testing model

In contrast to the problem of finding all defective elements in group testing, we consider the problem of finding one defective in a set D of defective elements of cardinality d. We consider adaptive search algorithms only. A similar problem for the classical and threshold models was solved in [1]. In the present paper we consider the additive testing model. We obtain an optimal answer in the problem of adaptive search of one defective element in this model.

Limits on Support Recovery With Probabilistic Models: An Information-Theoretic Framework

The support recovery problem consists of determining a sparse subset of a set of variables that is relevant in generating a set of observations, and arises in a diverse range of settings, such as compressive sensing, subset selection in regression, and group testing. In this paper, we take a unified approach to support recovery problems, considering general probabilistic models relating a sparse data vector to an observation vector. We study the information-theoretic limits of both exact and partial support recovery, taking a novel approach motivated by thresholding techniques in channel coding. We provide general achievability and converse bounds characterizing the trade-off between the error probability and number of measurements, and we specialize these to the linear, 1-bit, and group testing models. In several cases, our bounds not only provide matching scaling laws in the necessary and sufficient number of measurements, but also sharp thresholds with matching constant factors. Our approach has several advantages over previous approaches. For the achievability part, we obtain sharp thresholds under broader scalings of the sparsity level and other parameters (e.g., signal-to-noise ratio) compared with several previous works, and for the converse part, we not only provide conditions under which the error probability fails to vanish, but also conditions under which it tends to one.

Group Testing with Multiple Inhibitor Sets and Error-Tolerant and Its Decoding Algorithms.

In this article, we advance a new group testing model [Formula: see text] with multiple inhibitor sets and error-tolerant and propose decoding algorithms for it to identify all its positives by using [Formula: see text]-disjunct matrix. The decoding complexity for it is [Formula: see text], where [Formula: see text]. Moreover, we extend this new group testing to threshold group testing and give the threshold group testing model [Formula: see text] with multiple inhibitor sets and error-tolerant. By using [Formula: see text]-disjunct matrix, we propose its decoding algorithms for gap g = 0 and g > 0, respectively. Finally, we point out that the new group testing is the natural generalization for the clone model.

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

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.

Optimal Nested Test Plan for Combinatorial Quantitative Group Testing

We consider the quantitative group testing problem where the objective is to identify defective items in a given population based on results of tests performed on subsets of the population. Under the quantitative group testing model, the result of each test reveals the number of defective items in the tested group. The minimum number of tests achievable by nested test plans was established by Aigner and Schughart in 1985 within a minimax framework. The optimal nested test plan offering this performance, however, was not obtained. In this paper, we establish the optimal nested test plan in a closed form. This optimal nested test plan is also order optimal among all test plans as the population size approaches infinity. Using heavy-hitter detection as a case study, we show via simulation examples orders of magnitude improvement of the group testing approach over two prevailing sampling-based approaches in detection accuracy and counter consumption. Other applications include anomaly detection and wideband spectrum sensing in cognitive radio systems.

An Almost Unbiased Estimator in Group Testing with Errors in Inspection

The idea of pooling samples into pools as a cost effective method of screening individuals for the presence of a disease in a large population is discussed. Group testing was designed to reduce diagnostic cost. Testing population in pools also lower misclassification errors in low prevalence population. In this study we violate the assumption of homogeneity and perfect tests by investigating estimation problem in the presence of test errors. This is accomplished through Maximum Likelihood Estimation (MLE). The purpose of this study is to determine an analytical procedure for bias reduction in estimating population prevalence using group testing procedure in presence of tests errors. Specifically, we construct an almost unbiased estimator in pool-testing strategy in presence of test errors and compute the modified MLE of the prevalence of the population. For single stage procedures, with equal group sizes, we also propose a numerical method for bias correction which produces an almost unbiased estimator with errors. The existence of bias has been shown with the help of Taylor's expansion series, for group sizes greater than one. The indicator function with errors is used in the development of the model. A modified formula for bias correction has been analytically shown to reduce the bias of a group testing model. Also, the Fisher information and asymptotic variance has been shown to exist. We use MATLAB software for simulation and verification of the model. Then various tables are drawn to illustrate how the modified bias formula behaves for different values of sensitivities and specificities.

Asymptotics of Fingerprinting and Group Testing: Tight Bounds From Channel Capacities

In this work we consider the large-coalition asymptotics of various fingerprinting and group testing games, and derive explicit expressions for the capacities for each of these models. We do this both for simple decoders (fast but suboptimal) and for joint decoders (slow but optimal). For fingerprinting, we show that if the pirate strategy is known, the capacity often decreases linearly with the number of colluders, instead of quadratically as in the uninformed fingerprinting game. For many attacks the joint capacity is further shown to be strictly higher than the simple capacity. For group testing, we improve upon known results about the joint capacities, and derive new explicit asymptotics for the simple capacities. These show that existing simple group testing algorithms are suboptimal, and that simple decoders cannot asymptotically be as efficient as joint decoders. For the traditional group testing model, we show that the gap between the simple and joint capacities is a factor 1.44 for large numbers of defectives.

Semiquantitative Group Testing

We propose a novel group testing method, termed semi-quantitative group testing, motivated by a class of problems arising in genome screening experiments. Semi-quantitative group testing (SQGT) is a (possibly) non-binary pooling scheme that may be viewed as a concatenation of an adder channel and an integer-valued quantizer. In its full generality, SQGT may be viewed as a unifying framework for group testing, in the sense that most group testing models are special instances of SQGT. For the new testing scheme, we define the notion of SQ-disjunct and SQ-separable codes, representing generalizations of classical disjunct and separable codes. We describe several combinatorial and probabilistic constructions for such codes. While for most of these constructions we assume that the number of defectives is much smaller than total number of test subjects, we also consider the case in which there is no restriction on the number of defectives and they may be as large as the total number of subjects. For the codes constructed in this paper, we describe a number of efficient decoding algorithms. In addition, we describe a belief propagation decoder for sparse SQGT codes for which no other efficient decoder is currently known. Finally, we define the notion of capacity of SQGT and evaluate it for some special choices of parameters using information theoretic methods.

Strict group testing and the set basis problem

Group testing is the problem to identify up to d defectives out of n elements, by testing subsets for the presence of defectives. Let t(n,d,s) be the optimal number of tests needed by an s-stage strategy in the strict group testing model where the searcher must also verify that at most d defectives are present. We start building a combinatorial theory of strict group testing. We compute many exact t(n,d,s) values, thereby extending known results for s=1 to multistage strategies. These are interesting since asymptotically nearly optimal group testing is possible already in s=2 stages. Besides other combinatorial tools we generalize d-disjunct matrices to any candidate hypergraphs, and we reveal connections to the set basis problem and communication complexity. As a proof of concept we apply our tools to determine almost all test numbers for n≤10 and some further t(n,2,2) values. We also show t(n,2,2)≤2.44log2n+o(log2n).