具有可选择阈值的分组测试
Group Testing with Selectable Thresholds
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中文总结 AI 辅助
研究分组测试问题，引入可选择阈值模型，在最大阈值大或无界时建立高概率恢复条件及速率，固定最大阈值时确定可实现测试次数并得出逆命题，二者在密集极限下一致。
中文摘要 AI 辅助
我们考虑分组测试问题，即要从\(n\)个物品的更大集合中识别出大小为\(k\)的有缺陷物品子集，基于混合测试来进行。我们引入了一个可选择阈值模型，其中每个测试都有一个可选择的相关阈值，当且仅当测试中有缺陷物品的数量不小于该阈值时测试结果为\(1\)。在最大阈值大或无界的设置中，我们建立了条件，在此条件下以接近其最大可能值\(1\)的速率（即\(\log_2{n \choose k}\)与测试次数的渐近比率）可实现高概率恢复。此外，在固定最大阈值的情况下，我们使用简单且计算高效的解码方法确定了可实现的测试次数，并在测试设计的适当正则条件下得出了一个逆命题，在密集极限（即\(k = \Theta(n^{\theta})\)中\(\theta\)趋近于\(1\)）下两者一致。
英文摘要
We consider the problem of group testing, in which one seeks to identify a subset of defective items of size $k$ from a larger set of $n$ items based on pooled tests. We introduce a selectable threshold model, in which each test has an associated threshold that can be chosen, such that the test outcome is 1 if and only if the number of defectives in the test is no smaller than that threshold. In settings with a large or unbounded maximum threshold, we establish conditions under which high-probability recovery can be attained with a rate (i.e., the asymptotic ratio of $\log_2{n \choose k}$ to the number of tests) approaching its maximum possible value of 1. Moreover, in the case of a fixed maximum threshold, we establish an achievable number of tests using simple and computationally efficient decoding methods, and a converse that holds under suitable regularity conditions on the test design, with the two coinciding in the dense limit (i.e., $θ$ approaching one in the scaling $k = Θ(n^θ)$).
