On the Parallel 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sigma\in \{0, 1\}^{n}$</tex> , 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. For the regime where the signal is sparse such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Vert\sigma\Vert_{1}= o(n)$</tex> our results are twofold: 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. Our first result matches the performance guarantees of much more involved constructions (Karimi et al. 2019). Our second result extends a result of Alaoui et al. (2014) and Scarlett & Cevher (2017) who studied the pooled data problem for dense signals. Finally, our theoretical findings are complemented with empirical simulations. Our data not only confirm the information-theoretic thresholds but also hint at the practical applicability of our pooling scheme and the simple greedy reconstruction algorithm.