池化数据的并行重建
On the Parallel Reconstruction from Pooled Data
摘要：
在合并数据问题中，目标是从附加测量有效地重建二进制信号。给定一个信号$σ\in \{ 0,1 \}^n$，我们可以一次查询多个条目，并得到查询中非零条目的总数。我们假设查询是耗时的，因此关注于所有查询都并行执行的设置。对于信号稀疏的制度，使$ || σ||_1 = o(n)$我们的结果是双重的：首先，我们提出并分析了一个简单而有效的贪婪重建算法。其次，我们推导出一个尖锐的信息理论阈值的最小数量的查询重建$σ$具有高概率。我们的第一个结果与更复杂的构造的性能保证相匹配（Karimi et al. 2019）。我们的第二个结果扩展了Alaoui等人的结果。（2014）和Scarlett & Cevher（2017）研究了密集信号的合并数据问题。最后，我们的理论研究结果与实证模拟的补充。我们的数据不仅证实了信息理论阈值，但也暗示了我们的池化方案和简单的贪婪重建算法的实用性。
英文摘要：
In the pooled data problem the goal is to efficiently reconstruct a binary signal from additive measurements. Given a signal $σ\in \{ 0,1 \}^n$, 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 $ || σ||_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 $σ$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
