Performance bounds for LASSO under multiplicative LogNormal noise: Applications to pooled RT-PCR testing

Abstract

Group/Pooled testing is a technique which involves testing n samples for a rare disease in an indirect manner: instead of individually testing each of the p samples, it involves creating n<p pools and testing each pool, where a pool consists of a mixture of small, equal portions of a subset of the p samples. Group testing is known to save a large amount of testing time and resources in a variety of application settings. There also exist good theoretical guarantees for the recovery of the status of the p samples from results on n pools, if certain conditions are satisfied. The popularity of group testing for RT-PCR testing is also well-known. The noise in quantitative RT-PCR is known to follow a multiplicative and data-dependent model, which emerges from properties inherent to RT-PCR. In recent literature, the corresponding linear systems for inference of the health status of p samples from results on n pools, have been solved using the well-known Lasso estimator and its variants (Ghosh et al., 2021), which have been widely used in the compressed sensing literature. However, the Lasso has typically been 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 general technique proposed in Hunt et al. (2018) works well for establishing performance bounds for Lasso in Poisson inverse problems, we adapt it to handle sparse signal reconstruction from compressive measurements with multiplicative noise, as would appear in pooled RT-PCR testing. In particular, we present high probability performance bounds and data-dependent weights for the Lasso and a weighted version of the Lasso under multiplicative lognormal Noise. We also show numerical results on simulated pooled RT-PCR data to empirically validate our developed bounds.