In this work we study the fundamental limits of approximate recovery in the context of group testing. One of the most well-known, theoretically optimal, and easy to implement testing procedures is the non-adaptive Bernoulli group testing problem, where all tests are conducted in parallel, and each item is chosen to be part of any certain test independently with some fixed probability. In this setting, there is an observed gap between the number of tests above which recovery is information theoretically (IT) possible, and the number of tests required by the currently best known efficient algorithms to succeed. Often times such gaps are explained by a phase transition in the landscape of the solution space of the problem (an Overlap Gap Property phase transition). In this paper we seek to understand whether such a phenomenon takes place for Bernoulli group testing as well. Our main contributions are the following: (1) We provide first moment evidence that, perhaps surprisingly, such a phase transition does not take place throughout the regime for which recovery is IT possible. This fact suggests that the model is in fact amenable to local search algorithms ; (2) we prove the complete absence of "bad" local minima for a part of the "hard" regime, a fact which implies an improvement over known theoretical results on the performance of efficient algorithms for approximate recovery without false-negatives, and finally (3) we present extensive simulations that strongly suggest that a very simple local algorithm known as Glauber Dynamics does indeed succeed, and can be used to efficiently implement the well-known (theoretically optimal) Smallest Satisfying Set (SSS) estimator.
翻译:在这项工作中,我们研究了在群体测试背景下大致恢复的基本限度。最著名的、理论上最理想的和易于执行测试程序的一个最容易执行的测试问题是非适应性的Bernoulli群体测试问题,所有测试都是平行进行的,每个项目都是独立选择的任何特定测试的一部分,并有一些固定的概率。在这种环境下,从理论上看可能实现恢复的测试数量到目前最著名的高效算法所需要的测试数量之间有明显差距。在这种差距中,最著名的、理论上最理想的、最容易实施测试程序的一个原因是问题解决方案空间的横向过渡(超脱差距财产阶段过渡 ) 。 在本文中,我们力求了解这种现象是否同时进行,每个项目都是独立测试的一部分。我们的主要贡献如下:(1) 我们首先提供证据,也许令人惊讶的是,这种阶段过渡并非在可能实现恢复的政权中进行。这一事实表明,该模型事实上适合当地搜索算法(我们证明,对于“最坏的”本地迷你”的“最坏的”部分,对于“最坏的“最坏的”最坏的“最坏的”的算法,我们最后使用的一个事实意味着我们所知道的“最坏的“最坏的”的“最坏的”的“最坏的“最坏的”的“最精确的算法”的算法性能可以改进的“最坏的”的“最坏的”的“最坏的“最坏的算法”的”的“最后的”性,意味着是,意味着一个事实可以改进的“最坏的“最坏的“最坏的“最坏的”的算法”的”的“最坏的算法性能的“最坏的算法性能的“最能的”的最后的”的”的“最能说明的“最坏的“最能的”的”的”的“最坏的“最能的算法”最后的算法,意味着的“最能的算法性能的算法性能的“最坏的算法性能的算法性能可以说明,意味着的”最后的改进的改进的改进的进行。