Identification and Correction of Permutation Errors in Compressed Sensing-Based Group Testing

Compressed sensing, which involves reconstruction of sparse signals from an under-determined linear system, has been recently applied to problems in group testing to save on the number of tests administered during a pandemic or other resource-constrained scenarios. In practical group testing in time-constrained situations, the results of two different groups can sometimes be mistakenly exchanged by a technician. This is called ‘permutation noise’ and it presents challenges in determining the signal vector containing p health status values of the participating subjects from the results on n ≪ p pooled tests. In this paper, we present a method to determine the health status values in a manner that is robust to a small number of such permutations. The technique is based on a ‘debiased’ form of the robust LASSO estimator, with which we carefully design hypothesis tests in order (i) to identify the unhealthy subjects (based on non-zero values in the health status signal vector), and (ii) to identify the pooled measurements which were corrupted by permutation noise. Furthermore, we present an algorithm to correct the permutations in the pooled tests and subsequently reconstruct the signal vector from the corrected measurements. We further provide empirical results showing the efficacy of both the identification and correction of permutation errors, and show that it is superior to many intuitive baseline techniques.