scholarly journals Optimal group testing

2021 ◽  
pp. 1-38
Author(s):  
Amin Coja-Oghlan ◽  
Oliver Gebhard ◽  
Max Hahn-Klimroth ◽  
Philipp Loick
Keyword(s):  
Phase Transition ◽  
High Probability ◽  
Test Procedure ◽  
Group Testing ◽  
Test Design ◽  
Adaptive Scheme ◽  
Information Theoretic ◽  
Testing Problem ◽  
Small Set ◽  
Two Stages

Abstract In the group testing problem the aim is to identify a small set of k ⁓ n θ infected individuals out of a population size n, 0 < θ < 1. We avail ourselves of a test procedure capable of testing groups of individuals, with the test returning a positive result if and only if at least one individual in the group is infected. The aim is to devise a test design with as few tests as possible so that the set of infected individuals can be identified correctly with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition minf for non-adaptive group testing, where all tests are conducted in parallel. Thus with more than minf tests the infected individuals can be identified in polynomial time with high probability, while learning the set of infected individuals is information-theoretically impossible with fewer tests. In addition, we develop an optimal adaptive scheme where the tests are conducted in two stages.

BOUNDS FOR NONADAPTIVE GROUP TESTS TO ESTIMATE THE AMOUNT OF DEFECTIVES

2011 ◽  
Vol 03 (04) ◽  
pp. 517-536 ◽  
Author(s):  
PETER DAMASCHKE ◽  
AZAM SHEIKH MUHAMMAD
Keyword(s):  
Error Probability ◽  
Group Testing ◽  
Constant Factor ◽  
Constant Number ◽  
Constant Ratio ◽  
Information Theoretic ◽  
Small Constant ◽  
Test Number ◽  
Linear Programming Formulation ◽  
Two Stages

Group testing is the problem of finding d defectives in a set of n elements, by asking carefully chosen subsets (pools) whether they contain defectives. Strategies are preferred that use both a small number of tests close to the information-theoretic lower bound d log 2 n, and a small constant number of stages, where tests in every stage are done in parallel, in order to save time. They should even work if d is not known in advance. In fact, one can succeed with O(d log n) queries in two stages, if certain tests are randomized and a constant failure probability is allowed. An essential ingredient of such strategies is to get an estimate of d within a constant factor. This problem is also interesting in its own right. It can be solved with O( log n) randomized group tests of a certain type. We prove that Ω( log n) tests are also necessary, if elements for the pools are chosen independently. The proof builds upon an analysis of the influence of tests on the searcher's ability to distinguish between any two candidate numbers with a constant ratio. The next challenge is to get optimal constant factors in the O( log n) test number, depending on the prescribed error probability and the accuracy of d. We give practical methods to derive upper bound tradeoffs and conjecture that they are already close to optimal. One of them uses a linear programming formulation.

Quantum information entropy and squeezing of 𝒫𝒯-symmetric potential

2021 ◽  
Vol 36 (10) ◽  
pp. 2150065
Author(s):  
Aarti Sharma ◽  
Pooja Thakur ◽  
Girish Kumar ◽  
Anil Kumar
Keyword(s):  
Phase Transition ◽  
Quantum Information ◽  
Information Entropy ◽  
Momentum Space ◽  
Quantum Mechanical ◽  
Position Space ◽  
Entropy Squeezing ◽  
Information Theoretic ◽  
Symmetric Potential ◽  
Quantum Mechanical Systems

The information theoretic concepts are crucial to study the quantum mechanical systems. In this paper, the information densities of [Formula: see text]-symmetric potential have been demonstrated and their properties deeply analyzed. The position space and momentum space information entropy is obtained and Bialynicki-Birula–Mycielski inequality is saturated for different parameters of the potential. Some interesting features of information entropy have been discussed. The variation in these entropies is described which gets saturated for specific values of the parameter. These have also been analyzed for the [Formula: see text]-symmetry breaking case. Further, the entropy squeezing phenomenon has been investigated in position space as well as momentum space. Interestingly, [Formula: see text] phase transition conjectures the entropy squeezing in position space and momentum space.

scholarly journals Dorfman and R1-type procedures for a generalized group- testing problem

1972 ◽  
Vol 15 (3-4) ◽  
pp. 317-340 ◽  
Author(s):  
Jung-Keun Lee ◽  
Milton Sobel
Keyword(s):  
Group Testing ◽  
Testing Problem

scholarly journals Group Testing-Based Robust Algorithm for Diagnosis of COVID-19

Diagnostics ◽  
2020 ◽  
Vol 10 (6) ◽  
pp. 396
Author(s):  
Jin-Taek Seong
Keyword(s):  
Large Population ◽  
Group Testing ◽  
Incidence Rates ◽  
Posteriori Probability ◽  
Iterative Detection ◽  
Robust Algorithm ◽  
A Posteriori Probability ◽  
Small Set ◽  
Diagnosis Method ◽  
Severe Infections

At the time of writing, the COVID-19 infection is spreading rapidly. Currently, there is no vaccine or treatment, and researchers around the world are attempting to fight the infection. In this paper, we consider a diagnosis method for COVID-19, which is characterized by a very rapid rate of infection and is widespread. A possible method for avoiding severe infections is to stop the spread of the infection in advance by the prompt and accurate diagnosis of COVID-19. To this end, we exploit a group testing (GT) scheme, which is used to find a small set of confirmed cases out of a large population. For the accurate detection of false positives and negatives, we propose a robust algorithm (RA) based on the maximum a posteriori probability (MAP). The key idea of the proposed RA is to exploit iterative detection to propagate beliefs to neighbor nodes by exchanging marginal probabilities between input and output nodes. As a result, we show that our proposed RA provides the benefit of being robust against noise in the GT schemes. In addition, we demonstrate the performance of our proposal with a number of tests and successfully find a set of infected samples in both noiseless and noisy GT schemes with different COVID-19 incidence rates.

scholarly journals Near-optimal matrix recovery from random linear measurements

2018 ◽  
Vol 115 (28) ◽  
pp. 7200-7205 ◽  
Author(s):  
Elad Romanov ◽  
Matan Gavish
Keyword(s):  
Phase Transition ◽  
Message Passing ◽  
Iterative Solvers ◽  
Transition Curve ◽  
Linear Measurements ◽  
Information Theoretic ◽  
Matrix Estimation ◽  
Recovery Algorithm ◽  
Matrix Recovery ◽  
Approximate Message Passing

In matrix recovery from random linear measurements, one is interested in recovering an unknown M-by-N matrixX0fromn<MNmeasurementsyi=Tr(Ai⊤X0), where eachAiis an M-by-N measurement matrix with i.i.d. random entries,i=1,…,n. We present a matrix recovery algorithm, based on approximate message passing, which iteratively applies an optimal singular-value shrinker—a nonconvex nonlinearity tailored specifically for matrix estimation. Our algorithm typically converges exponentially fast, offering a significant speedup over previously suggested matrix recovery algorithms, such as iterative solvers for nuclear norm minimization (NNM). It is well known that there is a recovery tradeoff between the information content of the objectX0to be recovered (specifically, its matrix rank r) and the number of linear measurements n from which recovery is to be attempted. The precise tradeoff between r and n, beyond which recovery by a given algorithm becomes possible, traces the so-called phase transition curve of that algorithm in the(r,n)plane. The phase transition curve of our algorithm is noticeably better than that of NNM. Interestingly, it is close to the information-theoretic lower bound for the minimal number of measurements needed for matrix recovery, making it not only state of the art in terms of convergence rate, but also near optimal in terms of the matrices it successfully recovers.

scholarly journals On the Construction of Unbiased Estimators for the Group Testing Problem

Sankhya A ◽  
2018 ◽  
Vol 82 (1) ◽  
pp. 220-241
Author(s):  
Gregory Haber ◽  
Yaakov Malinovsky
Keyword(s):  
Group Testing ◽  
Testing Problem ◽  
Unbiased Estimators

TOPOLOGICAL MASS GENERATION, ELECTROWEAK SYMMETRY BREAKING AND BARYOGENESIS

2009 ◽  
Vol 24 (09) ◽  
pp. 703-711
Author(s):  
B. BASU ◽  
P. BANDYOPADHYAY
Keyword(s):  
Phase Transition ◽  
Symmetry Breaking ◽  
Electroweak Symmetry Breaking ◽  
Chiral Anomaly ◽  
Electroweak Theory ◽  
Electroweak Symmetry ◽  
Mass Generation ◽  
Two Stages ◽  
Topological Mass ◽  
Good Agreement

We have studied here electroweak symmetry breaking and baryogenesis from the viewpoint of topological mass generation through chiral anomaly. It is shown that the SU(2) gauge symmetry of the electroweak theory breaks in two stages. In the final stage we have Z-strings produced at the phase transition. We have also studied the problem of baryogenesis in this formalism and the ratio of the baryon–antibaryon is found to be in good agreement with the observed value.

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