# On the Parallel Reconstruction from Pooled Data

## TL;DR

This paper introduces a simple greedy algorithm for reconstructing sparse binary signals from pooled additive measurements in parallel, establishing sharp theoretical thresholds and validating them through empirical simulations.

## Contribution

It presents a new efficient greedy reconstruction algorithm and derives the exact information-theoretic query threshold for sparse signals in pooled data problems.

## Key findings

- The greedy algorithm achieves performance comparable to complex methods.
- Theoretical thresholds for minimal queries are established and validated.
- Empirical results confirm the practical effectiveness of the approach.

## Abstract

In the pooled data problem the goal is to efficiently reconstruct a binary signal from additive measurements. Given a signal $\sigma \in \{ 0,1 \}^n$, 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. For the regime where the signal is sparse such that $ || \sigma ||_1 = o(n)$ 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 $\sigma$ 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.