Matrix Completion in Group Testing: Bounds and Simulations

The main goal of group testing is to identify a small number of defective items in a large population of items. A test on a subset of items is positive if the subset contains at least one defective item and negative otherwise. In non-adaptive design, all tests can be tested simultaneously and represented by a measurement matrix in which a row and a column represent a test and an item, respectively. An entry in row $i$ and column $j$ is 1 if item $j$ belongs to the test $i$ and is 0 otherwise. Given an unknown set of defective items, the objective is to design a measurement matrix such that, by observing its corresponding outcome vector, the defective items can be recovered efficiently. The basic trait of this approach is that the measurement matrix has remained unchanged throughout the course of generating the outcome vector and recovering defective items. In this paper, we study the case in which some entries in the measurement matrix are erased, called \emph{the missing measurement matrix}, before the recovery phase of the defective items, and our objective is to fully recover the measurement matrix from the missing measurement matrix. In particular, we show that some specific rows with erased entries provide information aiding the recovery while others do not. Given measurement matrices and erased entries follow the Bernoulli distribution, we show that before the erasing event happens, sampling sufficient sets of defective items and their corresponding outcome vectors can help us recover the measurement matrix from the missing measurement matrix.

Concomitant Group Testing

In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).