Novel Impossibility Results for Group-Testing (1801.02701v3)

Abstract: In this work we prove non-trivial impossibility results for perhaps the simplest non-linear estimation problem, that of {\it Group Testing} (GT), via the recently developed Madiman-Tetali inequalities. Group Testing concerns itself with identifying a hidden set of $d$ defective items from a set of $n$ items via $t$ {disjunctive/pooled} measurements ("group tests"). We consider the linear sparsity regime, i.e. $d = \delta n$ for any constant $\delta >0$, a hitherto little-explored (though natural) regime. In a standard information-theoretic setting, where the tests are required to be non-adaptive and a small probability of reconstruction error is allowed, our lower bounds on $t$ are the {\it first} that improve over the classical counting lower bound, $t/n \geq H(\delta)$, where $H(\cdot)$ is the binary entropy function. As corollaries of our result, we show that (i) for $\delta \gtrsim 0.347$, individual testing is essentially optimal, i.e., $t \geq n(1-o(1))$; and (ii) there is an {adaptivity gap}, since for $\delta \in (0.3471,0.3819)$ known {adaptive} GT algorithms require fewer than $n$ tests to reconstruct ${\cal D}$, whereas our bounds imply that the best nonadaptive algorithm must essentially be individual testing of each element. Perhaps most importantly, our work provides a framework for combining combinatorial and information-theoretic methods for deriving non-trivial lower bounds for a variety of non-linear estimation problems.

• The paper establishes novel lower bounds for non-adaptive group testing in the linear sparsity regime using Madiman-Tetali inequalities, exceeding classical counting bounds.
• It reveals a notable adaptivity gap for certain parameter ranges, showing that nonadaptive testing may require significantly more tests than adaptive methods.
• The findings suggest that for specific sparsity levels, no nonadaptive strategy is substantially better than testing each item individually, highlighting limits of nonadaptive test efficacy.

Impossibility Results in Group Testing via Madiman-Tetali Inequalities

This paper presents significant novel impossibility results for the classical Group Testing (GT) problem, with a keen focus on the non-adaptive linear sparsity regime. Utilizing the Madiman-Tetali inequalities, the authors offer improved lower bounds for the number of tests required, addressing regions previously neglected by classical counting bounds. This insight provides a new viewpoint for the typically underexplored scenario where the number of defectives $d$ scales linearly with the total items $n$, specifically where $d = \Theta(n)$.

Overview of Group Testing

Group Testing revolves around identifying defective items from a larger set using pooled tests. A test indicates positivity if any defective item is present in the pool. The fundamental challenge is to minimize the number of tests, $t$, needed to accurately identify all defective items, allowing a small probability of reconstruction error. While prior research has extensively covered sub-linear sparsity regimes using adaptive and non-adaptive strategies, the linear sparsity regime has lacked rigorous paper.

Contributions and Key Results

The authors establish the first set of lower bounds on test numbers that exceed classical counting bounds for linear sparsity in a non-adaptive testing framework, which is pivotal for information-theoretic settings.

1. Improvement over Classical Bound:
   • For $\delta > 0.347$, the results demonstrate that individual testing becomes essentially optimal, aligning with $(1 - o(1))n$, revealing an intriguing adaptivity gap.
2. Adaptivity Gap:
   • A notable adaptivity gap is uncovered for $\delta \in (0.3471, 0.3819)$, where nonadaptive testing may require significantly more tests than adaptive methods.
3. Framework Development:
   • The novel framework combines combinatorial and information-theoretic techniques to bridge the gap in our understanding of non-linear estimation limits within GT.
4. Strong Numerical Insights:
   • By employing three distinct converse bounds, the paper effectively shows that for certain $\delta$ ranges, no better nonadaptive strategy exists than testing each item, marking a boundary in nonadaptive test efficacy.

Theoretical and Practical Implications

Theoretical implications of this work extend primarily to the domains of information theory and combinatorics, suggesting new potential inquiry avenues for improving suboptimal bounds. Practically, the findings can impact various applications such as bioinformatics, communication systems, and more, where GT problems are prevalent.

The results encourage revisiting adaptive methods and their perceived advantages, pushing for continuous exploration of test-design algorithms that could leverage identified inefficiencies in nonadaptive approaches.

Speculations on Future Developments

Presented findings, especially the demonstrated adaptivity gap, necessitate further investigation into more nuanced mechanisms that exploit adaptive versatility. Future pursuits may examine the application of these results across other sparse recovery frameworks—potentially extending to threshold group testing and secret-sharing scenarios.

Establishing adaptive strategies that outperform nonadaptive results for broader parameter spaces could substantially optimize GT configurations in practice. Additionally, the fusion of combinatorial and information-theoretic pursuits might inspire innovations within coding theory and its applications.

In summary, this paper breaks new ground in the paper of group-testing inefficiencies, charting a path for more refined models addressing non-linear estimation problems. The introduction of the Madiman-Tetali inequalities serves not only to advance theoretical understandings but also challenges researchers to explore more embedded structural complexities in similar estimation tasks.