Noisy group testing via spatial coupling

TL;DR

This work addresses noisy non-adaptive group testing for a sparse infected set $k=\lceil n^\theta\rceil$ within a population of size $n$, allowing general asymmetric noise in test outcomes. The authors introduce SPARC, a spatially coupled belief-propagation–inspired algorithm for approximate recovery, and SPEX for exact recovery, showing SPARC achieves $m\le(1+\varepsilon)c_{\mathrm{Sh}}k\ln(n/k)$ tests with $o(k)$ errors, while SPEX achieves exact recovery with $m\le(1+\varepsilon)c_{\mathrm{ex}}k\ln(n/k)$ tests, matching the information-theoretic lower bound on the constant-column design. A rigorous information-theoretic lower bound for the constant-column design is established, proving that no algorithm can achieve exact recovery with fewer than $m_{\mathrm{SPEX}}$ tests and, in the approximate-recovery regime, that $(1-\varepsilon)m_{\mathrm{SPARC}}$ tests are necessary. The analysis combines spatial coupling, a paced variant of BP, and detailed large-deviation arguments across multiple sections, and it provides explicit results for binary symmetric, Z-channel, and general asymmetric noise. The results show that even small noise dramatically alters optimal test designs and rates, highlighting the practical impact of robust, efficiently computable schemes in GT under realistic noise models.

Abstract

We study the problem of identifying a small set $k\sim n^θ$, $0<θ<1$, of infected individuals within a large population of size $n$ by testing groups of individuals simultaneously. All tests are conducted concurrently. The goal is to minimise the total number of tests required. In this paper we make the (realistic) assumption that tests are noisy, i.e.\ that a group that contains an infected individual may return a negative test result or one that does not contain an infected individual may return a positive test results with a certain probability. The noise need not be symmetric. We develop an algorithm called SPARC that correctly identifies the set of infected individuals up to $o(k)$ errors with high probability with the asymptotically minimum number of tests. Additionally, we develop an algorithm called SPEX that exactly identifies the set of infected individuals w.h.p. with a number of tests that matches the information-theoretic lower bound for the constant column design, a powerful and well-studied test design.

Figures (2)

• Figure 1: Information rates on different channels in nats. The horizontal axis displays the infection density parameter $0<\theta<1$. The colour indicates the optimal value of $d$ for a given $\theta$.
• Figure 2: The threshold function $\mathfrak z(\,\cdot\,)$ (red) on the interval $\mathcal{Y}(c_{\mathrm{ex},1}(d,\theta),d,\theta)$ and the resulting large deviations rate $c_{\mathrm{ex},1}(d,\theta)d(1-\theta)\left({D_{\mathrm{KL}}\left({{{\alpha}\|{\exp(-d)}}}\right)+\alpha D_{\mathrm{KL}}\left({{{\mathfrak z(\alpha)}\|{p_{01}}}}\right)}\right)$ (black) with $\theta=1/2$, $p_{00}=0.972$, $p_{11}=0.9$ at the optimal choice of $d$.

Theorems & Definitions (77)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5: Janson_2011
• Lemma 1.6: Hoeffding
• Theorem 2.1: Maurice
• Proposition 2.3
• Proposition 2.4: Maurice
• Proposition 2.5