Matrix Completion in Group Testing: Bounds and Simulations

TL;DR

This paper investigates the problem of reconstructing a measurement matrix in group testing when some entries are missing, providing theoretical bounds, complexity analysis, and simulation results that outperform existing methods.

Contribution

It introduces the matrix completion in group testing (MCGT) problem, proves NP-completeness, and develops bounds and algorithms validated through simulations.

Findings

NP-complete nature of MCGT problem

Missing entries can sometimes aid recovery

Larger s increases recovery probability

Abstract

The goal of group testing is to identify a small number of defective items within a large population. In the non-adaptive setting, tests are designed in advance and represented by a measurement matrix $\mM$, where rows correspond to tests and columns to items. A test is positive if it includes at least one defective item. Traditionally, $\mM$ remains fixed during both testing and recovery. In this work, we address the case where some entries of $\mM$ are missing, yielding a missing measurement matrix $\mG$. Our aim is to reconstruct $\mM$ from $\mG$ using available samples and their outcome vectors. The above problem can be considered as a problem intersected between Boolean matrix factorization and matrix completion, called the matrix completion in group testing (MCGT) problem, as follows. Given positive integers $t,s,n$, let $\mY:=(y_{ij}) \in \{0, 1\}^{t \times s}$, $\mM:=(m_{ij}) \in \{0,1\}^{t \times n}$, $\mX:=(x_{ij}) \in \{0,1\}^{n \times s}$, and matrix $\mG \in \{0,1 \}^{t \times n}$ be a matrix generated from matrix $\mM$ by erasing some entries in $\mM$. Suppose $\mY:=\mM \odot \mX$, where an entry $y_{ij}:=\bigvee_{k=1}^n (m_{ik}\wedge x_{kj})$, and $\wedge$ and $\vee$ are AND and OR operators. Unlike the problem in group testing whose objective is to find $\mX$ when given $\mM$ and $\mY$, our objective is to recover $\mM$ given $\mY,\mX$, and $\mG$. We first prove that the MCGT problem is NP-complete. Next, we show that certain rows with missing entries aid recovery while others do not. For Bernoulli measurement matrices, we establish that larger $s$ increases the higher the probability that $\mM$ can be recovered. We then instantiate our bounds for specific decoding algorithms and validate them through simulations, demonstrating superiority over standard matrix completion and Boolean matrix factorization methods.