50 research outputs found
Nearly Optimal Sparse Group Testing
Group testing is the process of pooling arbitrary subsets from a set of
items so as to identify, with a minimal number of tests, a "small" subset of
defective items. In "classical" non-adaptive group testing, it is known
that when is substantially smaller than , tests are
both information-theoretically necessary and sufficient to guarantee recovery
with high probability. Group testing schemes in the literature meeting this
bound require most items to be tested times, and most tests
to incorporate items.
Motivated by physical considerations, we study group testing models in which
the testing procedure is constrained to be "sparse". Specifically, we consider
(separately) scenarios in which (a) items are finitely divisible and hence may
participate in at most tests; or (b) tests are
size-constrained to pool no more than items per test. For both
scenarios we provide information-theoretic lower bounds on the number of tests
required to guarantee high probability recovery. In both scenarios we provide
both randomized constructions (under both -error and zero-error
reconstruction guarantees) and explicit constructions of designs with
computationally efficient reconstruction algorithms that require a number of
tests that are optimal up to constant or small polynomial factors in some
regimes of and . The randomized design/reconstruction
algorithm in the -sized test scenario is universal -- independent of the
value of , as long as . We also investigate the effect of
unreliability/noise in test outcomes. For the full abstract, please see the
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Group testing:an information theory perspective
The group testing problem concerns discovering a small number of defective
items within a large population by performing tests on pools of items. A test
is positive if the pool contains at least one defective, and negative if it
contains no defectives. This is a sparse inference problem with a combinatorial
flavour, with applications in medical testing, biology, telecommunications,
information technology, data science, and more. In this monograph, we survey
recent developments in the group testing problem from an information-theoretic
perspective. We cover several related developments: efficient algorithms with
practical storage and computation requirements, achievability bounds for
optimal decoding methods, and algorithm-independent converse bounds. We assess
the theoretical guarantees not only in terms of scaling laws, but also in terms
of the constant factors, leading to the notion of the {\em rate} of group
testing, indicating the amount of information learned per test. Considering
both noiseless and noisy settings, we identify several regimes where existing
algorithms are provably optimal or near-optimal, as well as regimes where there
remains greater potential for improvement. In addition, we survey results
concerning a number of variations on the standard group testing problem,
including partial recovery criteria, adaptive algorithms with a limited number
of stages, constrained test designs, and sublinear-time algorithms.Comment: Survey paper, 140 pages, 19 figures. To be published in Foundations
and Trends in Communications and Information Theor
Poisson Group Testing: A Probabilistic Model for Boolean Compressed Sensing
We introduce a novel probabilistic group testing framework, termed Poisson
group testing, in which the number of defectives follows a right-truncated
Poisson distribution. The Poisson model has a number of new applications,
including dynamic testing with diminishing relative rates of defectives. We
consider both nonadaptive and semi-adaptive identification methods. For
nonadaptive methods, we derive a lower bound on the number of tests required to
identify the defectives with a probability of error that asymptotically
converges to zero; in addition, we propose test matrix constructions for which
the number of tests closely matches the lower bound. For semi-adaptive methods,
we describe a lower bound on the expected number of tests required to identify
the defectives with zero error probability. In addition, we propose a
stage-wise reconstruction algorithm for which the expected number of tests is
only a constant factor away from the lower bound. The methods rely only on an
estimate of the average number of defectives, rather than on the individual
probabilities of subjects being defective
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Algorithms to Exploit Data Sparsity
While data in the real world is very high-dimensional, it generally has some underlying structure; for instance, if we think of an image as a set of pixels with associated color values, most possible settings of color values correspond to something more like random noise than what we typically think of as a picture. With an appropriate transformation of basis, this underlying structure can often be converted into sparsity in data, giving an equivalent representation of the data where the magnitude is large in only a few directions relative to the ambient dimension. This motivates a variety of theoretical questions around designing algorithms that can exploit this data sparsity to achieve better performance than what would be possible naively, and in this thesis we tackle several such questions.We first examine the question of simply approximating the level of sparsity of a signal under several different measurement models, a natural first step if the sparsity is to be exploited by other algorithms. Second, we look at a particular sparse signal recovery problem called nonadaptive probabilistic group testing, and investigate the question of exactly how sparse the signal needs to be before the methods used for recovering sparse signals outperform those used for non-sparse signals. Third, we prove novel upper bounds on the number of measurements needed to recover a sparse signal in the universal one-bit compressed sensing model of sparse signal recovery. Fourth, we give some approximations of an information-theoretic quantity called the index coding rate of a network modeled by a graph, in the special case that the graph is sparse or otherwise highly structured. For each of the problems considered, we also discuss some remaining open questions and conjectures, as well as possible directions towards their solutions
Noisy Non-Adaptive Group Testing: A (Near-)Definite Defectives Approach
The group testing problem consists of determining a small set of defective
items from a larger set of items based on a number of possibly-noisy tests, and
is relevant in applications such as medical testing, communication protocols,
pattern matching, and many more. We study the noisy version of the problem,
where the output of each standard noiseless group test is subject to
independent noise, corresponding to passing the noiseless result through a
binary channel. We introduce a class of algorithms that we refer to as
Near-Definite Defectives (NDD), and study bounds on the required number of
tests for vanishing error probability under Bernoulli random test designs. In
addition, we study algorithm-independent converse results, giving lower bounds
on the required number of tests under Bernoulli test designs. Under reverse
-channel noise, the achievable rates and converse results match in a broad
range of sparsity regimes, and under -channel noise, the two match in a
narrower range of dense/low-noise regimes. We observe that although these two
channels have the same Shannon capacity when viewed as a communication channel,
they can behave quite differently when it comes to group testing. Finally, we
extend our analysis of these noise models to the symmetric noise model, and
show improvements over the best known existing bounds in broad scaling regimes.Comment: Submitted to IEEE Transactions on Information Theor
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
Engineering Competitive and Query-Optimal Minimal-Adaptive Randomized Group Testing Strategies
Suppose that given is a collection of elements where 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 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 and requires only queries. Usually competitive group testing is based on sequential queries. We have shown that actually competitive group testing with expected queries is possible in only or 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 -stage randomized strategy with prescribed success probability can asymptotically achieve the information-theoretic lower bound for and growing much slower than . Similarly, we can approach the entropy lower bound in stages when