Problem
Research questions and friction points this paper is trying to address.
Threshold Group Testing
exact recovery
sparse binary vector
non-adaptive testing
information-theoretic threshold
Algorithms for Threshold Group Testing
🤖 AI Summary
This work addresses the problem of exact recovery of sparse binary vectors in noiseless, non-adaptive threshold group testing, where test outcomes indicate whether the number of defective items exceeds a predetermined threshold. The authors propose an efficient inference algorithm based on spatially coupled test designs, which achieves exact recovery with high probability using a constant-column-weight test matrix and a number of tests matching the information-theoretic optimum. By introducing a simplified algorithmic framework that circumvents intricate weighted-sum analyses, the approach significantly reduces theoretical complexity while maintaining near-optimal performance guarantees. The optimality and efficiency of the method are further corroborated through information-theoretic phase transition analysis.
📝 Abstract
We study the Threshold Group Testing (TGT) problem without a gap in the noiseless, non-adaptive setting, where the goal is to exactly recover a sparse binary vector from pooled test outcomes using as few tests as possible. In TGT, a test applied to a subset of items returns a positive outcome if the number of defective items in the subset reaches a prescribed threshold, and a negative outcome otherwise. Under the assumption of an analytic condition, TGT has been shown to undergo a sharp information-theoretic phase transition for exact recovery on the class of constant-column test designs. In this paper, we develop an efficient inference algorithm that achieves exact recovery with high probability using the minimum number of non-adaptive tests that are needed for the constant-column design, thereby matching the information-theoretic threshold of a natural benchmark test design. Our approach is based on a spatially coupled test design and admits a significantly simpler analysis than existing algorithms for related group testing problems. In particular, unlike previous methods for binary group testing, our algorithm does not rely on the analysis of intricate weighted sums. This leads to a more straightforward proof technique, while still allowing near-optimal performance guarantees.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Threshold Group Testing
spatial coupling
non-adaptive testing
exact recovery
information-theoretic threshold
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