线性区域下采用次线性测试的定量分组检测与混合数据
Quantitative Group Testing and Pooled Data in the Linear Regime with Sublinear Tests
AI总结：
该研究针对非自适应线性区域下的定量分组检测与混合数据问题，提出基于空间耦合伯努利测试矩阵和AMP算法的恢复方案，证明其无噪下次线性测试几乎精确恢复，且噪声下性能平稳下降，有限维下优于现有方案。
摘要：
在混合数据问题中，研究目标是通过一系列混合测试识别大量物品对应的类别。每次混合测试会揭示池中属于每个类别的物品数量。一个突出的特例是定量分组检测（QGT），即两类别的混合数据情况。我们在非自适应和线性区域下研究这些问题，其中每个类别的物品占比为常数阶。我们提出一种采用空间耦合伯努利测试矩阵的方案，以及一种用于恢复的高效近似消息传递（AMP）算法。我们严格刻画了其在无噪声和有噪声场景下的渐近性能，并证明在无噪声情况下，AMP算法仅需总物品数$p$的次线性数量级的测试，即可实现几乎精确的恢复。尽管已有其他针对无噪声QGT和混合数据的高效方案能以阶最优的样本复杂度（$\Theta(\frac{p}{\log p})$次测试）实现恢复，但这些方案在存在噪声（即使是低噪声水平）时的性能没有任何保证。相比之下，我们的方案在无噪声情况下仅需$p$的次线性数量级的测试即可实现恢复，且在存在噪声时性能会平稳下降。数值仿真验证了空间耦合方案在有限维度下的优势，结果表明其性能优于独立同分布（i.i.d.）测试设计以及其他基于凸规划的恢复算法。
英文摘要：
In the pooled data problem, the goal is to identify the categories associated with a large collection of items via a sequence of pooled tests. Each pooled test reveals the number of items in the pool belonging to each category. A prominent special case is quantitative group testing (QGT), which is the case of pooled data with two categories. We consider these problems in the non-adaptive and linear regime, where the fraction of items in each category is of constant order. We propose a scheme with a spatially coupled Bernoulli test matrix and an efficient approximate message passing (AMP) algorithm for recovery. We rigorously characterize its asymptotic performance in both the noiseless and noisy settings, and prove that in the noiseless case, the AMP algorithm achieves almost-exact recovery with a number of tests sublinear in the total number of items $p$. Although there exist other efficient schemes for noiseless QGT and pooled data that achieve recovery with order-optimal sample complexity ($Θ(\frac{p}{\log p})$ tests), there are no guarantees on their performance in the presence of noise, even at low noise-levels. In comparison, our scheme achieves recovery in the noiseless case with a number of tests sublinear in $p$, and its performance degrades gracefully in the presence of noise. Numerical simulations illustrate the benefits of the spatially coupled scheme at finite dimensions, showing that it outperforms i.i.d. test designs as well as other recovery algorithms based on convex programming.
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