4 research outputs found
Estimation of Sparsity via Simple Measurements
We consider several related problems of estimating the 'sparsity' or number
of nonzero elements in a length vector by observing only
, where is a predesigned test matrix
independent of , and the operation varies between problems.
We aim to provide a -approximation of sparsity for some constant
with a minimal number of measurements (rows of ). 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 rows are sufficient
when the operation is logical OR (i.e., group testing), and nearly this
many are necessary, where is a known upper bound on . When instead the
operation is multiplication over or a finite field
, we show that respectively and measurements are necessary and sufficient.Comment: 13 pages; shortened version presented at ISIT 201
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 \emph{defective} elements out of elements by querying subsets (pools) for the presence of defectives.
By the information-theoretic lower bound, essentially 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 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 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 and show that actually competitive GT is possible in as few as stages only.
The main ingredient is our randomized estimate of a previously unknown using nonadaptive queries.
In addition, we have developed a systematic approach to obtain optimal competitive ratios for our strategies.
When is a known upper bound,
we propose randomized GT strategies which asymptotically achieve query optimality in just , or stages depending upon the growth of versus .
Inspired by application settings, such as at American Red Cross, where in most cases GT is applied to small instances, \textit{e.g.}, . We extended our study of query-optimal GT strategies to solve a given problem instance with fixed values , and . 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 tests are sufficient, and also necessary for certain ranges of the parameters