关于伯努利群检测的全有或全无行为
On the All-Or-Nothing Behavior of Bernoulli Group Testing
AI总结：
本文研究非自适应群检测在弱恢复准则下的极限，发现伯努利随机测试存在全有或全无现象：测试数略低于阈值时弱恢复不可能，略高于时精确恢复可行，而非伯努利设计可用更少测试实现所有放宽准则。
摘要：
本文研究了非自适应群检测问题，在该问题中，人们试图根据一组设计得当的测试结果来识别哪些物品是有缺陷的，这些测试的结果指示测试中是否至少包含一个有缺陷的物品。最广泛的恢复准则旨在精确恢复整个缺陷集合，同时也有考虑诸如近似恢复和列表解码等放宽的准则。在本文中，我们研究了在显著放宽的弱恢复准则下群检测的基本极限，该准则仅旨在识别一小部分（例如，0.01）的有缺陷物品。鉴于在足够稀疏的缩放机制下，独立同分布伯努利测试对于精确恢复具有接近最优性，自然要问这种设计在弱恢复下是否也能以更少的测试成功。我们的主要负面结果表明情况并非如此，事实上，在足够稀疏机制下的独立同分布伯努利随机测试中，会出现一种全有或全无现象：当测试数量略低于某个阈值时，弱恢复是不可能的，而当测试数量略高于同一阈值时，高概率的精确恢复是可能的。在建立这一结果的过程中，我们还证明了在伯努利设计下，对于弱检测问题（区分群检测模型与完全随机结果）以及识别单个确定有缺陷物品的问题，存在类似的负面结果。在积极方面，我们表明，在适当选择的非伯努利设计下，所有三个放宽的恢复准则都可以用相当少的测试来实现。
英文摘要：
In this paper, we study the problem of non-adaptive group testing, in which one seeks to identify which items are defective given a set of suitably-designed tests whose outcomes indicate whether or not at least one defective item was included in the test. The most widespread recovery criterion seeks to exactly recover the entire defective set, and relaxed criteria such as approximate recovery and list decoding have also been considered. In this paper, we study the fundamental limits of group testing under the significantly relaxed {\em weak recovery} criterion, which only seeks to identify a small fraction (e.g., $0.01$) of the defective items. Given the near-optimality of i.i.d.~Bernoulli testing for exact recovery in sufficiently sparse scaling regimes, it is natural to ask whether this design additionally succeeds with much fewer tests under weak recovery. Our main negative result shows that this is not the case, and in fact, under i.i.d.~Bernoulli random testing in the sufficiently sparse regime, an {\em all-or-nothing} phenomenon occurs: When the number of tests is slightly below a threshold, weak recovery is impossible, whereas when the number of tests is slightly above the same threshold, high-probability exact recovery is possible. In establishing this result, we additionally prove similar negative results under Bernoulli designs for the weak detection problem (distinguishing between the group testing model vs.~completely random outcomes) and the problem of identifying a single item that is definitely defective. On the positive side, we show that all three relaxed recovery criteria can be attained using considerably fewer tests under suitably-chosen non-Bernoulli designs.
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