4,457 research outputs found
Near-Optimal Noisy Group Testing via Separate Decoding of Items
The group testing problem consists of determining a small set of defective
items from a larger set of items based on a number of tests, and is relevant in
applications such as medical testing, communication protocols, pattern
matching, and more. In this paper, we revisit an efficient algorithm for noisy
group testing in which each item is decoded separately (Malyutov and Mateev,
1980), and develop novel performance guarantees via an information-theoretic
framework for general noise models. For the special cases of no noise and
symmetric noise, we find that the asymptotic number of tests required for
vanishing error probability is within a factor of the
information-theoretic optimum at low sparsity levels, and that with a small
fraction of allowed incorrectly decoded items, this guarantee extends to all
sublinear sparsity levels. In addition, we provide a converse bound showing
that if one tries to move slightly beyond our low-sparsity achievability
threshold using separate decoding of items and i.i.d. randomized testing, the
average number of items decoded incorrectly approaches that of a trivial
decoder.Comment: Submitted to IEEE Journal of Selected Topics in Signal Processin
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
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
full text PDF
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
Non-adaptive Group Testing: Explicit bounds and novel algorithms
We consider some computationally efficient and provably correct algorithms
with near-optimal sample-complexity for the problem of noisy non-adaptive group
testing. Group testing involves grouping arbitrary subsets of items into pools.
Each pool is then tested to identify the defective items, which are usually
assumed to be "sparse". We consider non-adaptive randomly pooling measurements,
where pools are selected randomly and independently of the test outcomes. We
also consider a model where noisy measurements allow for both some false
negative and some false positive test outcomes (and also allow for asymmetric
noise, and activation noise). We consider three classes of algorithms for the
group testing problem (we call them specifically the "Coupon Collector
Algorithm", the "Column Matching Algorithms", and the "LP Decoding Algorithms"
-- the last two classes of algorithms (versions of some of which had been
considered before in the literature) were inspired by corresponding algorithms
in the Compressive Sensing literature. The second and third of these algorithms
have several flavours, dealing separately with the noiseless and noisy
measurement scenarios. Our contribution is novel analysis to derive explicit
sample-complexity bounds -- with all constants expressly computed -- for these
algorithms as a function of the desired error probability; the noise
parameters; the number of items; and the size of the defective set (or an upper
bound on it). We also compare the bounds to information-theoretic lower bounds
for sample complexity based on Fano's inequality and show that the upper and
lower bounds are equal up to an explicitly computable universal constant factor
(independent of problem parameters).Comment: Accepted for publication in the IEEE Transactions on Information
Theory; current version, Oct. 9, 2012. Main change from v4 to v5: fixed some
typos, corrected details of the LP decoding algorithm
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