Estimation of Sparsity via Simple Measurements

We consider several related problems of estimating the 'sparsity' or number of nonzero elements $d$ in a length $n$ vector $\mathbf{x}$ by observing only $\mathbf{b} = M \odot \mathbf{x}$, where $M$ is a predesigned test matrix independent of $\mathbf{x}$, and the operation $\odot$ varies between problems. We aim to provide a $\Delta$-approximation of sparsity for some constant $\Delta$ with a minimal number of measurements (rows of $M$). This framework generalizes multiple problems, such as estimation of sparsity in group testing and compressed sensing. We use techniques from coding theory as well as probabilistic methods to show that $O(D \log D \log n)$ rows are sufficient when the operation $\odot$ is logical OR (i.e., group testing), and nearly this many are necessary, where $D$ is a known upper bound on $d$. When instead the operation $\odot$ is multiplication over $\mathbb{R}$ or a finite field $\mathbb{F}_q$, we show that respectively $\Theta(D)$ and $\Theta(D \log_q \frac{n}{D})$ measurements are necessary and sufficient.Comment: 13 pages; shortened version presented at ISIT 201

Engineering Competitive and Query-Optimal Minimal-Adaptive Randomized Group Testing Strategies

Suppose that given is a collection of $n$ elements where $d$ of them are \emph{defective}. We can query an arbitrarily chosen subset of elements which returns Yes if the subset contains at least one defective and No if the subset is free of defectives. The problem of group testing is to identify the defectives with a minimum number of such queries. By the information-theoretic lower bound at least $\log_2 \binom {n}{d} \approx d\log_2 (\frac{n}{d}) \approx d\log_2 n$ queries are needed. Using adaptive group testing, i.e., asking one query at a time, the lower bound can be easily achieved. However, strategies are preferred that work in a fixed small number of stages, where queries in a stage are asked in parallel. A group testing strategy is called \emph{competitive} if it works for completely unknown $d$ and requires only $O(d\log_2 n)$ queries. Usually competitive group testing is based on sequential queries. We have shown that actually competitive group testing with expected $O(d\log_2 n)$ queries is possible in only $2$ or $3$ stages. Then we have focused on minimizing the hidden constant factor in the query number and proposed a systematic approach for this purpose. Another main result is related to the design of query-optimal and minimal-adaptive strategies. We have shown that a $2$-stage randomized strategy with prescribed success probability can asymptotically achieve the information-theoretic lower bound for $d \ll n$ and growing much slower than $n$. Similarly, we can approach the entropy lower bound in $4$ stages when $d=o(n)$

New Constructions for Competitive and Minimal-Adaptive Group Testing

Group testing (GT) was originally proposed during the World War II in an attempt to minimize the \emph{cost} and \emph{waiting time} in performing identical blood tests of the soldiers for a low-prevalence disease. Formally, the GT problem asks to find $d\ll n$ \emph{defective} elements out of $n$ elements by querying subsets (pools) for the presence of defectives. By the information-theoretic lower bound, essentially $d\log_2 n$ queries are needed in the worst-case. An \emph{adaptive} strategy proceeds sequentially by performing one query at a time, and it can achieve the lower bound. In various applications, nothing is known about $d$ beforehand and a strategy for this scenario is called \emph{competitive}. Such strategies are usually adaptive and achieve query optimality within a constant factor called the \emph{competitive ratio}. In many applications, queries are time-consuming. Therefore, \emph{minimal-adaptive} strategies which run in a small number $s$ of stages of parallel queries are favorable. This work is mainly devoted to the design of minimal-adaptive strategies combined with other demands of both theoretical and practical interest. First we target unknown $d$ and show that actually competitive GT is possible in as few as $2$ stages only. The main ingredient is our randomized estimate of a previously unknown $d$ using nonadaptive queries. In addition, we have developed a systematic approach to obtain optimal competitive ratios for our strategies. When $d$ is a known upper bound, we propose randomized GT strategies which asymptotically achieve query optimality in just $2$, $3$ or $4$ stages depending upon the growth of $d$ versus $n$. Inspired by application settings, such as at American Red Cross, where in most cases GT is applied to small instances, \textit{e.g.}, $n=16$. We extended our study of query-optimal GT strategies to solve a given problem instance with fixed values $n$, $d$ and $s$. We also considered the situation when elements to test cannot be divided physically (electronic devices), thus the pools must be disjoint. For GT with \emph{disjoint} simultaneous pools, we show that $\Theta (sd(n/d)^{1/s})$ tests are sufficient, and also necessary for certain ranges of the parameters