GROTESQUE：噪声组测试（快速高效）
GROTESQUE: Noisy Group Testing (Quick and Efficient)
摘要：
分组测试是指通过“少量”“合并”测试从（大量）$N$项目中识别（高概率）$D$缺陷的（小）子集的问题。为了便于介绍，在这项工作中，我们专注于政权时，$D = \cO{N^{1-\gap}}$的一些$\gap > 0$。测试可以是无噪声的或有噪声的，并且测试过程可以是自适应的（定义测试的池可以取决于先前测试的结果）或非自适应的（每个测试独立于其他测试的结果来执行）。大量的文献表明，$Θ(D\log(N))$测试的信息理论上是必要的和充分的组测试问题，并提供算法，实现这一性能。然而，直到最近才开始研究$N$中具有次线性计算复杂度的重建算法（\cite{GurI:04,IndN:10, NgoP:11}最近的工作给出了一些第一个这样的算法）。在这种情况下，自适应测试与嘈杂的结果，我们提出了第一个计划，同时订单最佳（小常数因子）的测试和解码复杂性（$\cO{D\log(N)}$在这两个性能指标）的数量。我们的自适应算法的阶段的总数是“小”（$\cO{\log(D)}$）。类似地，在具有噪声结果的非自适应测试的场景中，我们提出了在测试数量和解码复杂度两者中同时接近最优的第一方案（通过需要$\cO{D\log(D)\log(N)}$测试并且具有{ ${\cal O}(D(\log N+\log^{2}D))$ }的解码复杂度的算法）。最后，我们提出了一个自适应算法，只需要2个阶段，测试的数量和解码复杂度的规模为{ ${\cal O}(D(\log N+\log^{2}D))$ }。对于所有三种设置，我们的算法的错误概率按比例缩放为$\cO{1/(poly(D)}$。
英文摘要：
Group-testing refers to the problem of identifying (with high probability) a (small) subset of $D$ defectives from a (large) set of $N$ items via a "small" number of "pooled" tests. For ease of presentation in this work we focus on the regime when $D = \cO{N^{1-\gap}}$ for some $\gap > 0$. The tests may be noiseless or noisy, and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or non-adaptive (each test is performed independent of the outcome of other tests). A rich body of literature demonstrates that $Θ(D\log(N))$ tests are information-theoretically necessary and sufficient for the group-testing problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexity that is sub-linear in $N$ have started being investigated (recent work by \cite{GurI:04,IndN:10, NgoP:11} gave some of the first such algorithms). In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously order-optimal (up to small constant factors) in both the number of tests and the decoding complexity ($\cO{D\log(N)}$ in both the performance metrics). The total number of stages of our adaptive algorithm is "small" ($\cO{\log(D)}$). Similarly, in the scenario with non-adaptive tests with noisy outcomes, we present the first scheme that is simultaneously near-optimal in both the number of tests and the decoding complexity (via an algorithm that requires $\cO{D\log(D)\log(N)}$ tests and has a decoding complexity of {${\cal O}(D(\log N+\log^{2}D))$}. Finally, we present an adaptive algorithm that only requires 2 stages, and for which both the number of tests and the decoding complexity scale as {${\cal O}(D(\log N+\log^{2}D))$}. For all three settings the probability of error of our algorithms scales as $\cO{1/(poly(D)}$.
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。
