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Noisy Linear Group Testing: Exact Thresholds and Efficient Algorithms

Lukas Hintze, Lena Krieg, Olga Scheftelowitsch, Haodong Zhu

TL;DR

This work considers the setting where each item is defective with a constant probability $\alpha$, independent of all other items, and provides efficient algorithms that identify all defective items with the optimal amount of tests with high probability.

Abstract

In group testing, the task is to identify defective items by testing groups of them together using as few tests as possible. We consider the setting where each item is defective with a constant probability $α$, independent of all other items. In the (over-)idealized noiseless setting, tests are positive exactly if any of the tested items are defective. We study a more realistic model in which observed test results are subject to noise, i.e., tests can display false positive or false negative results with constant positive probabilities. We determine precise constants $c$ such that $cn\log n$ tests are required to recover the infection status of every individual for both adaptive and non-adaptive group testing: in the former, the selection of groups to test can depend on previously observed test results, whereas it cannot in the latter. Additionally, for both settings, we provide efficient algorithms that identify all defective items with the optimal amount of tests with high probability. Thus, we completely solve the problem of binary noisy group testing in the studied setting.

Noisy Linear Group Testing: Exact Thresholds and Efficient Algorithms

TL;DR

This work considers the setting where each item is defective with a constant probability , independent of all other items, and provides efficient algorithms that identify all defective items with the optimal amount of tests with high probability.

Abstract

In group testing, the task is to identify defective items by testing groups of them together using as few tests as possible. We consider the setting where each item is defective with a constant probability , independent of all other items. In the (over-)idealized noiseless setting, tests are positive exactly if any of the tested items are defective. We study a more realistic model in which observed test results are subject to noise, i.e., tests can display false positive or false negative results with constant positive probabilities. We determine precise constants such that tests are required to recover the infection status of every individual for both adaptive and non-adaptive group testing: in the former, the selection of groups to test can depend on previously observed test results, whereas it cannot in the latter. Additionally, for both settings, we provide efficient algorithms that identify all defective items with the optimal amount of tests with high probability. Thus, we completely solve the problem of binary noisy group testing in the studied setting.

Paper Structure

This paper contains 44 sections, 54 theorems, 140 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related work
  4. Model and Results
  5. Model and commonly used notation
  6. Results
  7. Preliminaries
  8. MAP estimation
  9. Notation used throughout
  10. Non-adaptive Group Testing
  11. Impossibility: proof strategy for \ref{['thm:nalower']}
  12. The algorithm $\texttt{SPOG}$: proof strategy for \ref{['thm:naalg']}
  13. Adaptive Group Testing
  14. Impossibility: proof strategy for \ref{['thm:adp:low']}
  15. The algorithm $\texttt{PRESTO}$: proof strategy for \ref{['thm:adp:upper']}
  16. ...and 29 more sections

Key Result

Theorem 1

For any valid noisy channel $\bm{p}$, $\alpha>0$ and $\varepsilon>0$, there exists some $n_0 = n_0(\bm{p}, \alpha, \varepsilon)$ such that for every $n>n_0$, all test designs $G$ with $m<(1-\varepsilon)m_{\mathrm{na}} (\alpha, \bm{p})$ tests and every estimation function $f_G:\left\{{0,1}\right\}^m

Figures (3)

  • Figure 1: Comparison of adaptive and non-adaptive thresholds $c_{\text{ad}}=m_{\mathrm{ad}} /(n\ln n)$ and $c_{\text{na}}=m_{\mathrm{na}} /(n\ln n)$.
  • Figure 2: Illustration of $\bm{G}_{\texttt{SPOG}}$. Circles represent individuals and squares tests.
  • Figure 3: An illustration of the adaptive test scheme of PRESTO. The circles represent the individuals and the squares represent tests.

Theorems & Definitions (94)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 3.1: MAP estimate
  • Definition 3.3: good tests
  • Definition 4.1: Genie estimator
  • Lemma 4.2: Genie estimator is better than MAP estimator
  • Lemma 4.3
  • Lemma 4.4
  • ...and 84 more