Noisy Nonadaptive Group Testing with Binary Splitting: New Test Design and Improvement on Price-Scarlett-Tan's Scheme

Abstract: In Group Testing, the objective is to identify $K$ defective items out of $N$, $K\ll N$, by testing pools of items together and using the least amount of tests possible. Recently, a fast decoding method based on binary splitting (Price and Scarlett, 2020) has been proposed that simultaneously achieve optimal number of tests and decoding complexity for Non-Adaptive Probabilistic Group Testing (NAPGT). However, the method works only when the test results are noiseless. In this paper, we further study the binary splitting method and propose (1) A NAPGT scheme that generalizes the original binary splitting method from the noiseless case into tests with $\rho$ proportion of false positives (the $\rho$-False Positive Channel), where $\rho$ is a constant, with asymptotically-optimal number of tests and decoding complexity, i.e. $\mathcal{O}(K\log N)$, and (2) A NAPGT scheme in the presence of both false positives and false negatives in test outcomes, improving and generalizing the work of Price, Scarlett and Tan~\cite{price2023fast} in two ways: First, under $\rho$-proportion of test results flipped ($\rho$-Binary Symmetric Channel) and within the general sublinear regime $K=\Theta(N^\alpha)$ where $0<\alpha<1$, our algorithm has a decoding complexity of $\mathcal{O}(\epsilon^{-2}K^{1+\epsilon})$ where $\epsilon>0$ is a constant parameter. Second, when the false negative flipping probability $\rho'$ satisfies $\rho'=\mathcal{O}(K^{-\epsilon})$ and the false positive flipping probability $\rho$ is a constant, we can simultaneously achieve $\mathcal{O}(\epsilon^{-1}K\log N)$ for both the number of tests and the decoding complexity. It remains open to achieve these optimals under the general BSC.

• The paper extends Price and Scarlett's noiseless algorithm to noisy settings by introducing the NEON scheme that meets the optimal O(K log N) bounds.
• It employs a two-phase approach with Saffron-based initial decoding and false negative correction to effectively handle errors from Binary Asymmetric and Binary Symmetric Channels.
• The research offers practical solutions for large-scale applications like medical diagnostics and network communication and sets the stage for future advancements in group testing methods.

Nonadaptive Noisy Group Testing with Optimal Tests and Decoding

The paper presents a significant advancement in the field of group testing (GT) by extending Price and Scarlett's algorithm to accommodate noisy environments, specifically addressing the Binary Asymmetric Channel (BAC) and Binary Symmetric Channel (BSC). The proposed scheme achieves optimal performance in both test efficiency and decoding complexity, a notable milestone in probabilistic nonadaptive group testing (NAPGT) under noise.

Key Contributions

The research develops a nonadaptive probabilistic group testing framework that efficiently identifies defective items among a larger set, achieving the theoretical lower bound of $O(K\log N)$ for both the number of tests and decoding complexity. By generalizing the work of Price and Scarlett from noiseless scenarios to those with binary noise, the authors address a longstanding challenge in group testing.

Methodology

The authors introduce the NEON (Noise-resilient, Efficient, and Optimal testiNg) scheme, which leverages local decoding to effectively manage false positive errors, while incorporating a novel strategy for correcting false negatives. The algorithm operates in two primary phases:

1. Initial Decoding with Saffron: The scheme employs Saffron-based group testing to identify a subset of defectives even in the presence of both false positive and negative errors. This initial phase provides a preliminary set of defective items, which then assists in correcting the false negatives.
2. False Negative Correction and Final Decoding: Using the initial results, the scheme corrects tests known to include defectives but appear negative due to noise. This correction significantly enhances the reliability of the final decoding, allowing the optimal decoding complexity to be reached.

Numerical Results and Comparisons

The proposed NEON scheme is benchmarked against existing methods, demonstrating superior performance in achieving error probabilities that diminish as $N$ and $K$ increase. The comparative analysis focuses on scenarios with various sparsity levels, parameterized by $\alpha$, and noise characteristics, achieving robustness across a wide spectrum of conditions.

Practical and Theoretical Implications

Practically, this work holds promise for applications in large-scale testing scenarios such as medical diagnostics or network communication, where minimal testing and computational overhead are crucial. Theoretically, the research provides a framework for future endeavors aiming to bridge the gap between test efficiency and decoding complexity in noisy environments.

Future Directions

While the current work addresses $K=o(N^{1-\frac{2(1+\eta)}{C})$, further research might explore extending these results to cover the entire sublinear regime. Additionally, refining the algorithm to handle higher noise probabilities or developing an error probability model independent of $K$ could offer further advancements.

In conclusion, this paper makes a vital contribution to the field of group testing by demonstrating how optimal bounds can be maintained even under noisy conditions, thus advancing both theoretical understanding and practical application of NAPGT methods.