Information-theoretic and algorithmic aspects of parallel and distributed reconstruction from pooled data

Abstract

In the pooled data problem the goal is to efficiently reconstruct a binary signal from additive measurements. Given a signal σ∈{0,1}n, we can query multiple entries at once and get the total number of non-zero entries in the query as a result. We assume that queries are time-consuming and therefore focus on the setting where all queries are executed in parallel. First, we propose and analyze a simple and efficient greedy reconstruction algorithm. Secondly, we derive a sharp information-theoretic threshold for the minimum number of queries required to reconstruct σ with high probability. Finally, we consider two noise models: In the noisy channel model, the result for each entry of the signal flips with a certain probability. In the noisy query model, each query result is subject to random Gaussian noise. We pin down the range of error probabilities and distributions for which our algorithm reconstructs the exact initial states with high probability. Our theoretical findings are complemented by simulations where we compare our simple algorithm with approximate message passing (AMP) that is conjectured to be optimal in a number of related problems.