20,412 research outputs found
Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)
During the last two decades, concentration inequalities have been the subject
of exciting developments in various areas, including convex geometry,
functional analysis, statistical physics, high-dimensional statistics, pure and
applied probability theory, information theory, theoretical computer science,
and learning theory. This monograph focuses on some of the key modern
mathematical tools that are used for the derivation of concentration
inequalities, on their links to information theory, and on their various
applications to communications and coding. In addition to being a survey, this
monograph also includes various new recent results derived by the authors. The
first part of the monograph introduces classical concentration inequalities for
martingales, as well as some recent refinements and extensions. The power and
versatility of the martingale approach is exemplified in the context of codes
defined on graphs and iterative decoding algorithms, as well as codes for
wireless communication. The second part of the monograph introduces the entropy
method, an information-theoretic technique for deriving concentration
inequalities. The basic ingredients of the entropy method are discussed first
in the context of logarithmic Sobolev inequalities, which underlie the
so-called functional approach to concentration of measure, and then from a
complementary information-theoretic viewpoint based on transportation-cost
inequalities and probability in metric spaces. Some representative results on
concentration for dependent random variables are briefly summarized, with
emphasis on their connections to the entropy method. Finally, we discuss
several applications of the entropy method to problems in communications and
coding, including strong converses, empirical distributions of good channel
codes, and an information-theoretic converse for concentration of measure.Comment: Foundations and Trends in Communications and Information Theory, vol.
10, no 1-2, pp. 1-248, 2013. Second edition was published in October 2014.
ISBN to printed book: 978-1-60198-906-
Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
One of the key issues in the acquisition of sparse data by means of
compressed sensing (CS) is the design of the measurement matrix. Gaussian
matrices have been proven to be information-theoretically optimal in terms of
minimizing the required number of measurements for sparse recovery. In this
paper we provide a new approach for the analysis of the restricted isometry
constant (RIC) of finite dimensional Gaussian measurement matrices. The
proposed method relies on the exact distributions of the extreme eigenvalues
for Wishart matrices. First, we derive the probability that the restricted
isometry property is satisfied for a given sufficient recovery condition on the
RIC, and propose a probabilistic framework to study both the symmetric and
asymmetric RICs. Then, we analyze the recovery of compressible signals in noise
through the statistical characterization of stability and robustness. The
presented framework determines limits on various sparse recovery algorithms for
finite size problems. In particular, it provides a tight lower bound on the
maximum sparsity order of the acquired data allowing signal recovery with a
given target probability. Also, we derive simple approximations for the RICs
based on the Tracy-Widom distribution.Comment: 11 pages, 6 figures, accepted for publication in IEEE transactions on
information theor
Aspects of generic entanglement
We study entanglement and other correlation properties of random states in
high-dimensional bipartite systems. These correlations are quantified by
parameters that are subject to the "concentration of measure" phenomenon,
meaning that on a large-probability set these parameters are close to their
expectation. For the entropy of entanglement, this has the counterintuitive
consequence that there exist large subspaces in which all pure states are close
to maximally entangled. This, in turn, implies the existence of mixed states
with entanglement of formation near that of a maximally entangled state, but
with negligible quantum mutual information and, therefore, negligible
distillable entanglement, secret key, and common randomness. It also implies a
very strong locking effect for the entanglement of formation: its value can
jump from maximal to near zero by tracing over a number of qubits negligible
compared to the size of total system. Furthermore, such properties are generic.
Similar phenomena are observed for random multiparty states, leading us to
speculate on the possibility that the theory of entanglement is much simplified
when restricted to asymptotically generic states. Further consequences of our
results include a complete derandomization of the protocol for universal
superdense coding of quantum states.Comment: 22 pages, 1 figure, 1 tabl
Quantum data hiding in the presence of noise
When classical or quantum information is broadcast to separate receivers,
there exist codes that encrypt the encoded data such that the receivers cannot
recover it when performing local operations and classical communication, but
they can decode reliably if they bring their systems together and perform a
collective measurement. This phenomenon is known as quantum data hiding and
hitherto has been studied under the assumption that noise does not affect the
encoded systems. With the aim of applying the quantum data hiding effect in
practical scenarios, here we define the data-hiding capacity for hiding
classical information using a quantum channel. Using this notion, we establish
a regularized upper bound on the data hiding capacity of any quantum broadcast
channel, and we prove that coherent-state encodings have a strong limitation on
their data hiding rates. We then prove a lower bound on the data hiding
capacity of channels that map the maximally mixed state to the maximally mixed
state (we call these channels "mictodiactic"---they can be seen as a
generalization of unital channels when the input and output spaces are not
necessarily isomorphic) and argue how to extend this bound to generic channels
and to more than two receivers.Comment: 12 pages, accepted for publication in IEEE Transactions on
Information Theor
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