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Correction of Pooling Matrix Mis-specifications in Compressed Sensing Based Group Testing

Shuvayan Banerjee, Radhendushka Srivastava, James Saunderson, Ajit Rajwade

TL;DR

This work addresses correcting model-mismatch errors (MMEs) in real-valued compressed sensing-based group testing by estimating the unknown pooling matrix from observed results and the known design. It introduces a multi-stage Cape framework that detects MMEs with a debiased robust Lasso test (Odrlt), corrects them, and re-estimates the signal with Robust Lasso, backed by stability and reconstruction guarantees. The approach yields improved reconstruction accuracy and detection performance across multiple MME models (SSM, ASM, permutations) and noise regimes, including log-normal PCR-like noise, and benefits from fast, closed-form debiasing for common Bernoulli designs. The method requires no re-testing or new pools and demonstrates practical viability for resource-constrained screening with uncertain pooling operations, providing theoretical and empirical validation of its effectiveness.

Abstract

Compressed sensing, which involves the reconstruction of sparse signals from an under-determined linear system, has been recently used to solve problems in group testing. In a public health context, group testing aims to determine the health status values of p subjects from n<<p pooled tests, where a pool is defined as a mixture of small, equal-volume portions of the samples of a subset of subjects. This approach saves on the number of tests administered in pandemics or other resource-constrained scenarios. In practical group testing in time-constrained situations, a technician can inadvertently make a small number of errors during pool preparation, which leads to errors in the pooling matrix, which we term `model mismatch errors' (MMEs). This poses difficulties while determining health status values of the participating subjects from the results on n<<p pooled tests. In this paper, we present an algorithm to correct the MMEs in the pooled tests directly from the pooled results and the available (inaccurate) pooling matrix. Our approach then reconstructs the signal vector from the corrected pooling matrix, in order to determine the health status of the subjects. We further provide theoretical guarantees for the correction of the MMEs and the reconstruction error from the corrected pooling matrix. We also provide several supporting numerical results.

Correction of Pooling Matrix Mis-specifications in Compressed Sensing Based Group Testing

TL;DR

This work addresses correcting model-mismatch errors (MMEs) in real-valued compressed sensing-based group testing by estimating the unknown pooling matrix from observed results and the known design. It introduces a multi-stage Cape framework that detects MMEs with a debiased robust Lasso test (Odrlt), corrects them, and re-estimates the signal with Robust Lasso, backed by stability and reconstruction guarantees. The approach yields improved reconstruction accuracy and detection performance across multiple MME models (SSM, ASM, permutations) and noise regimes, including log-normal PCR-like noise, and benefits from fast, closed-form debiasing for common Bernoulli designs. The method requires no re-testing or new pools and demonstrates practical viability for resource-constrained screening with uncertain pooling operations, providing theoretical and empirical validation of its effectiveness.

Abstract

Compressed sensing, which involves the reconstruction of sparse signals from an under-determined linear system, has been recently used to solve problems in group testing. In a public health context, group testing aims to determine the health status values of p subjects from n<<p pooled tests, where a pool is defined as a mixture of small, equal-volume portions of the samples of a subset of subjects. This approach saves on the number of tests administered in pandemics or other resource-constrained scenarios. In practical group testing in time-constrained situations, a technician can inadvertently make a small number of errors during pool preparation, which leads to errors in the pooling matrix, which we term `model mismatch errors' (MMEs). This poses difficulties while determining health status values of the participating subjects from the results on n<<p pooled tests. In this paper, we present an algorithm to correct the MMEs in the pooled tests directly from the pooled results and the available (inaccurate) pooling matrix. Our approach then reconstructs the signal vector from the corrected pooling matrix, in order to determine the health status of the subjects. We further provide theoretical guarantees for the correction of the MMEs and the reconstruction error from the corrected pooling matrix. We also provide several supporting numerical results.
Paper Structure (24 sections, 2 theorems, 16 equations, 6 figures, 3 tables, 3 algorithms)

This paper contains 24 sections, 2 theorems, 16 equations, 6 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Group Testing
  3. Errors in Group Membership Specifications
  4. Existing Methods for Detecting/ Correcting MMEs
  5. Contributions of this paper
  6. Techniques for Correction of MMEs in Compressed Sensing based Group Testing
  7. Problem Setup
  8. Algorithm: Correction of MMEs
  9. Case 1: All MME-affected rows are known
  10. Case 2: MME-affected rows are unknown
  11. Identification of MMEs using Odrlt
  12. Algorithm: Discarding MMEs
  13. Multiple stages of correction
  14. Theoretical Results
  15. Stability of MME correction
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\boldsymbol{B} \in \mathbb{R}^{n \times p}$ be the known pooling matrix with i.i.d. entries drawn from a Bernoulli($\theta$) distribution, and let $\boldsymbol{\tilde{B}}$ denote an unknown corrupted version of $\boldsymbol{B}$ due to model mismatch errors (MMEs). For each $i \in [n]$, let $\bo $\blacksquare$

Figures (6)

  • Figure 1: Average sensitivity and specificity (100 noise realizations; fixed $\boldsymbol{\beta}^*$, $\boldsymbol{A}$, $\boldsymbol{\delta}^*$) for detecting non-zero entries of $\boldsymbol{\beta}^*$ using Cape, RL, Mmer, and Odrlt in the presence of SSM MMEs. $\boldsymbol{A}$ is $CB(0.1)$. Panels correspond to (EA), (EB), (EC), (ED).
  • Figure 2: Average sensitivity and specificity (100 noise realizations; fixed $\boldsymbol{\beta}^*$, $\boldsymbol{A}$, $\boldsymbol{\delta}^*$) for detecting non-zero entries of $\boldsymbol{\beta}^*$ using Cape, RL, Mmer, and Odrlt under SSM MMEs. $\boldsymbol{A}$ is $CB(0.5)$. Panels correspond to (EA), (EB), (EC), (ED).
  • Figure 3: Average RRMSE (100 noise realizations) under SSM mismatches for $\theta=0.1$. Methods: Cape, RL, Mmer. Panels correspond to (EA), (EB), (EC), (ED).
  • Figure 4: Average RRMSE (100 noise realizations) under SSM mismatches for $\theta=0.5$. Methods: Cape, RL, Mmer. Panels correspond to (EA), (EB), (EC), (ED).
  • Figure 5: Convergence of Stopping functions: Convergence of stopping function $f_{ape}$ for correction algorithm Cape for permutations errors. Each line in the plot represents the respective function values at all $10$ stages of correction for different numbers of permutation errors given by $r \in \{0,2,4,6,8,10\}$. The fixed parameters are $p=100,n=80,f_{sp}=0.05,f_{\sigma}=0.01$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1: Effective MMEs
  • Definition 2: Model-based Correctable Effective MMEs
  • Theorem 1
  • Theorem 2