Boolean Compressed Sensing and Noisy Group Testing (0907.1061v6)

Abstract: The fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests (measurements). The key contribution of this paper is in adopting a new information-theoretic perspective on group testing problems. We formulate the group testing problem as a channel coding/decoding problem and derive a single-letter characterization for the total number of tests used to identify the defective set. Although the focus of this paper is primarily on group testing, our main result is generally applicable to other compressive sensing models. The single letter characterization is shown to be order-wise tight for many interesting noisy group testing scenarios. Specifically, we consider an additive Bernoulli($q$) noise model where we show that, for $N$ items and $K$ defectives, the number of tests $T$ is $O(\frac{K\log N}{1-q})$ for arbitrarily small average error probability and $O(\frac{K^2\log N}{1-q})$ for a worst case error criterion. We also consider dilution effects whereby a defective item in a positive pool might get diluted with probability $u$ and potentially missed. In this case, it is shown that $T$ is $O(\frac{K\log N}{(1-u)^2})$ and $O(\frac{K^2\log N}{(1-u)^2})$ for the average and the worst case error criteria, respectively. Furthermore, our bounds allow us to verify existing known bounds for noiseless group testing including the deterministic noise-free case and approximate reconstruction with bounded distortion. Our proof of achievability is based on random coding and the analysis of a Maximum Likelihood Detector, and our information theoretic lower bound is based on Fano's inequality.

• The paper proposes an information-theoretic characterization of the number of tests needed, deriving order-wise tight scaling laws under various noise models.
• It shifts from traditional combinatorial approaches to using Shannon coding theory and mutual information bounds for additive Bernoulli and dilution noise models.
• The results emphasize that robust pooling designs and error-correcting strategies are essential for efficient sparse signal recovery in noisy environments.

Analysis of Boolean Compressed Sensing and Noisy Group Testing

The paper by George Kamal Atia and Venkatesh Saligrama presents an in-depth exploration of group testing through an information-theoretic lens, establishing novel connections between group testing and channel coding problems. The authors' methodology shifts from the traditional combinatorial design approach to an analysis rooted in Shannon coding theory, providing a framework applicable to varied noisy group testing and compressed sensing models.

Key Contributions

Primarily, the paper proposes a new information-theoretic characterization for the total number of tests required to identify defective items within a large set, deriving a single-letter expression for this purpose. Notably, the derived expression is demonstrably order-wise tight for various noisy group testing scenarios. The authors discussed an additive Bernoulli noise model and its effects on tests, showing that the number of tests $T$ scales as $O\left(\frac{K \log N}{1-q}\right)$ for a worst-case error metric. Additionally, the dilution model, which simulates the probability of a defective item being missed, requires tests to scale as $O\left(\frac{K \log N}{(1-u)^2}\right)$.

Numerical Results and Implications

The paper provides a comprehensive analysis of scaling laws in both noise-free and noisy scenarios, delineating conditions under which group testing can recover defective item sets with either small average error probability or zero worst-case error probability. Importantly, the authors present results indicating that both additive and dilution noise results in significant increases in the number of required tests, with dilution leading to a more pronounced effect. The results imply that as noise conditions worsen, intelligent pooling designs and error-correcting strategies become crucial.

The information-theoretic lower bounds, derived using Fano's inequality, complement their achievability results, confirming these scaling laws' tightness. This work showcases that careful application of mutual information bounds enables significant insights into the trade-offs necessary for practical deployment scenarios.

Theoretical and Practical Implications

Theoretically, the paper advances the understanding of sparse signal recovery, extending beyond binary testing to broader compressed sensing models, thus bridging a gap in noisy group testing literature. Practically, these insights are crucial for scenarios where the number of error-prone tests must be minimized. Applications such as disease screening, quality control, and cognitive radio systems stand to benefit considerably, saving resources while ensuring target identification's fidelity.

Future Directions

Potential future directions may include investigating other noise models, expanding to multi-stage testing procedures, and integrating more sophisticated recovery algorithms. Exploring adaptive designs where pooling strategies evolve based on intermediate outcomes could further optimize test efficiencies.

In conclusion, Atia and Saligrama's work introduces a new vista in group testing research, with implications across sparse recovery problems. Their information-theoretic perspective unlocks new methodologies for handling noise, paving the way for resource-efficient and robust testing regimes in large-scale applications.