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Truong, L.V., Aldridge, M. orcid.org/0000-0002-9347-1586 and Scarlett, J. (2020) On the All-or-Nothing Behavior of Bernoulli Group Testing. IEEE Journal on Selected Areas in Information Theory, 1 (3). pp. 669-680. ISSN 2641-8770
Abstract
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. |
Keywords: | Group testing, hypothesis testing, approximate recovery, phase transitions |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 17 Oct 2024 09:12 |
Last Modified: | 08 Nov 2024 11:44 |
Status: | Published |
Publisher: | Institute of Electrical and Electronics Engineers (IEEE) |
Identification Number: | 10.1109/jsait.2020.3039790 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:218506 |
Available Versions of this Item
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On the All-Or-Nothing Behavior of Bernoulli Group Testing. (deposited 08 Nov 2024 11:42)
- On the All-or-Nothing Behavior of Bernoulli Group Testing. (deposited 17 Oct 2024 09:12) [Currently Displayed]