Distributed Reconstruction of Noisy Pooled Data

In the pooled data problem we are given a set of $n$ agents, each of which holds a hidden state bit, either $0$ or $1$. A querying procedure returns for a query set the sum of the states of the queried agents. The goal is to reconstruct the states using as few queries as possible. In this paper we consider two noise models for the pooled data problem. In the noisy channel model, the result for each agent flips with a certain probability. In the noisy query model, each query result is subject to random Gaussian noise. Our results are twofold. First, we present and analyze for both error models a simple and efficient distributed algorithm that reconstructs the initial states in a greedy fashion. Our novel analysis pins down the range of error probabilities and distributions for which our algorithm reconstructs the exact initial states with high probability. Secondly, we present simulation results of our algorithm and compare its performance with approximate message passing (AMP) algorithms that are conjectured to be optimal in a number of related problems.