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The Noisy Quantitative Group Testing Problem

Tenghao Li, Neha Sangwan, Xiaxin Li, Arya Mazumdar

TL;DR

This work analyzes quantitative group testing under three observation models—noiseless, additive Gaussian noise, and noisy Z-channel—using a non-adaptive random design. It develops and compares two decoding strategies, a correlation-based linear estimator and a least-squares estimator (LSE), deriving both achievability and converse bounds on the number of tests $m$ needed for exact recovery. Notably, for Gaussian QGT the paper provides matching upper and lower bounds on sample complexity, establishing a tight characterization up to constants, while also detailing explicit bounds for the noiseless and Z-channel settings. The results quantify the performance gap between polynomial-time linear decoding and information-theoretically optimal decoding across noise regimes and offer non-asymptotic guarantees with concrete constants, informing practical test design and decoding strategies in noisy quantitative group testing.

Abstract

In this paper, we study the problem of quantitative group testing (QGT) and analyze the performance of three models: the noiseless model, the additive Gaussian noise model, and the noisy Z-channel model. For each model, we analyze two algorithmic approaches: a linear estimator based on correlation scores, and a least squares estimator (LSE). We derive upper bounds on the number of tests required for exact recovery with vanishing error probability, and complement these results with information-theoretic lower bounds. In the additive Gaussian noise setting, our lower and upper bounds match in order.

The Noisy Quantitative Group Testing Problem

TL;DR

This work analyzes quantitative group testing under three observation models—noiseless, additive Gaussian noise, and noisy Z-channel—using a non-adaptive random design. It develops and compares two decoding strategies, a correlation-based linear estimator and a least-squares estimator (LSE), deriving both achievability and converse bounds on the number of tests needed for exact recovery. Notably, for Gaussian QGT the paper provides matching upper and lower bounds on sample complexity, establishing a tight characterization up to constants, while also detailing explicit bounds for the noiseless and Z-channel settings. The results quantify the performance gap between polynomial-time linear decoding and information-theoretically optimal decoding across noise regimes and offer non-asymptotic guarantees with concrete constants, informing practical test design and decoding strategies in noisy quantitative group testing.

Abstract

In this paper, we study the problem of quantitative group testing (QGT) and analyze the performance of three models: the noiseless model, the additive Gaussian noise model, and the noisy Z-channel model. For each model, we analyze two algorithmic approaches: a linear estimator based on correlation scores, and a least squares estimator (LSE). We derive upper bounds on the number of tests required for exact recovery with vanishing error probability, and complement these results with information-theoretic lower bounds. In the additive Gaussian noise setting, our lower and upper bounds match in order.
Paper Structure (31 sections, 6 theorems, 110 equations, 1 table)

This paper contains 31 sections, 6 theorems, 110 equations, 1 table.

Table of Contents

  1. Introduction
  2. Noiseless Model
  3. Additive Gaussian Noise Model
  4. Noisy Z-Channel
  5. Other Noise Models
  6. Our Contributions
  7. Preliminaries
  8. Basic Notations
  9. Quantitative Group Testing
  10. Observation Models
  11. Noiseless model
  12. Additive Gaussian noise model
  13. Noisy Z-channel model
  14. Decoding methods
  15. Linear estimator (correlation-based decoder)
  16. ...and 16 more sections

Key Result

Theorem 1

Let $n,k,A,x^*,y$ be as defined in Section sec:preliminaries. In the noiseless model $y = Ax^*$, consider the linear decoder that outputs indices of the $k$ largest coordinates of $A^\top y$, the number of tests required to recover $x^*$ with vanishing error probability satisfies

Theorems & Definitions (16)

  • Theorem 1: Noiseless QGT: Linear Estimator
  • Theorem 2: Additive Gaussian QGT
  • Theorem 3: Noisy Z-channel QGT
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 6 more