Group testing is a technique which avoids individually testing $n$ samples for a rare disease and instead tests $n < p$ pools, where a pool consists of a mixture of small, equal portions of a subset of the $p$ samples. Group testing saves testing time and resources in many applications, including RT-PCR, with guarantees for the recovery of the status of the $p$ samples from results on $n$ pools. The noise in quantitative RT- PCR is inherently known to follow a multiplicative data-dependent model. In recent literature, the corresponding linear systems for inferring the health status of $p$ samples from results on $n$ pools have been solved using the Lasso estimator and its variants, which have been typically used in additive Gaussian noise settings. There is no existing literature which establishes performance bounds for Lasso for the multiplicative noise model associated with RT-PCR. After noting that a recent general technique, Hunt et al., works for Poisson inverse problems, we adapt it to handle sparse signal reconstruction from compressive measurements with multiplicative noise: we present high probability performance bounds and data-dependent weights for the Lasso and its weighted version. We also show numerical results on simulated pooled RT-PCR data to empirically validate our bounds.

翻译：群组检测是一种避免对$n$个样本逐一检测罕见疾病的技术，转而检测$n<p$个混合样本（每个混合样本由$p$个样本中某个子集的小等份混合而成）。该技术在包括RT-PCR在内的诸多应用中节省检测时间与资源，并能从$n$个混合样本的结果中保证恢复$p$个样本的状态。定量RT-PCR中的噪声本质已知遵循乘性数据依赖模型。近期文献中，用于从$n$个混合样本结果推断$p$个样本健康状况的相应线性系统已通过Lasso估计器及其变体求解，但这些方法通常适用于加性高斯噪声场景。目前尚无文献建立针对RT-PCR乘性噪声模型的Lasso性能界。鉴于近期Hunt等人提出的一般性技术适用于泊松逆问题，我们将其改进以处理基于乘性噪声的压缩感知稀疏信号重构：我们给出了Lasso及其加权版本的高概率性能界与数据依赖权重。我们还通过模拟混合RT-PCR数据的数值实验结果，实证验证了所提界的有效性。