Phase Transitions in the Pooled Data Problem
This addresses a fundamental problem in group testing and information theory with applications in medical testing and data analysis, providing rigorous theoretical bounds.
The paper studies the pooled data problem of identifying item labels from pooled test counts, establishing an exact asymptotic threshold for noiseless optimal decoding with a phase transition between complete success and failure, and shows that noise significantly increases test requirements even at low levels.
In this paper, we study the pooled data problem of identifying the labels associated with a large collection of items, based on a sequence of pooled tests revealing the counts of each label within the pool. In the noiseless setting, we identify an exact asymptotic threshold on the required number of tests with optimal decoding, and prove a phase transition between complete success and complete failure. In addition, we present a novel noisy variation of the problem, and provide an information-theoretic framework for characterizing the required number of tests for general random noise models. Our results reveal that noise can make the problem considerably more difficult, with strict increases in the scaling laws even at low noise levels. Finally, we demonstrate similar behavior in an approximate recovery setting, where a given number of errors is allowed in the decoded labels.