72 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-
Fundamental limits of many-user MAC with finite payloads and fading
Consider a (multiple-access) wireless communication system where users are
connected to a unique base station over a shared-spectrum radio links. Each
user has a fixed number of bits to send to the base station, and his signal
gets attenuated by a random channel gain (quasi-static fading). In this paper
we consider the many-user asymptotics of Chen-Chen-Guo'2017, where the number
of users grows linearly with the blocklength. In addition, we adopt a per-user
probability of error criterion of Polyanskiy'2017 (as opposed to classical
joint-error probability criterion). Under these two settings we derive bounds
on the optimal required energy-per-bit for reliable multi-access communication.
We confirm the curious behaviour (previously observed for non-fading MAC) of
the possibility of perfect multi-user interference cancellation for user
densities below a critical threshold. Further we demonstrate the suboptimality
of standard solutions such as orthogonalization (i.e., TDMA/FDMA) and treating
interference as noise (i.e. pseudo-random CDMA without multi-user detection).Comment: 38 pages, conference version accepted to IEEE ISIT 201
Algebraic Properties of Polar Codes From a New Polynomial Formalism
Polar codes form a very powerful family of codes with a low complexity
decoding algorithm that attain many information theoretic limits in error
correction and source coding. These codes are closely related to Reed-Muller
codes because both can be described with the same algebraic formalism, namely
they are generated by evaluations of monomials. However, finding the right set
of generating monomials for a polar code which optimises the decoding
performances is a hard task and channel dependent. The purpose of this paper is
to reveal some universal properties of these monomials. We will namely prove
that there is a way to define a nontrivial (partial) order on monomials so that
the monomials generating a polar code devised fo a binary-input symmetric
channel always form a decreasing set.
This property turns out to have rather deep consequences on the structure of
the polar code. Indeed, the permutation group of a decreasing monomial code
contains a large group called lower triangular affine group. Furthermore, the
codewords of minimum weight correspond exactly to the orbits of the minimum
weight codewords that are obtained from (evaluations) of monomials of the
generating set. In particular, it gives an efficient way of counting the number
of minimum weight codewords of a decreasing monomial code and henceforth of a
polar code.Comment: 14 pages * A reference to the work of Bernhard Geiger has been added
(arXiv:1506.05231) * Lemma 3 has been changed a little bit in order to prove
that Proposition 7.1 in arXiv:1506.05231 holds for any binary input symmetric
channe
Sparse graph codes for compression, sensing, and secrecy
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from student PDF version of thesis.Includes bibliographical references (p. 201-212).Sparse graph codes were first introduced by Gallager over 40 years ago. Over the last two decades, such codes have been the subject of intense research, and capacity approaching sparse graph codes with low complexity encoding and decoding algorithms have been designed for many channels. Motivated by the success of sparse graph codes for channel coding, we explore the use of sparse graph codes for four other problems related to compression, sensing, and security. First, we construct locally encodable and decodable source codes for a simple class of sources. Local encodability refers to the property that when the original source data changes slightly, the compression produced by the source code can be updated easily. Local decodability refers to the property that a single source symbol can be recovered without having to decode the entire source block. Second, we analyze a simple message-passing algorithm for compressed sensing recovery, and show that our algorithm provides a nontrivial f1/f1 guarantee. We also show that very sparse matrices and matrices whose entries must be either 0 or 1 have poor performance with respect to the restricted isometry property for the f2 norm. Third, we analyze the performance of a special class of sparse graph codes, LDPC codes, for the problem of quantizing a uniformly random bit string under Hamming distortion. We show that LDPC codes can come arbitrarily close to the rate-distortion bound using an optimal quantizer. This is a special case of a general result showing a duality between lossy source coding and channel coding-if we ignore computational complexity, then good channel codes are automatically good lossy source codes. We also prove a lower bound on the average degree of vertices in an LDPC code as a function of the gap to the rate-distortion bound. Finally, we construct efficient, capacity-achieving codes for the wiretap channel, a model of communication that allows one to provide information-theoretic, rather than computational, security guarantees. Our main results include the introduction of a new security critertion which is an information-theoretic analog of semantic security, the construction of capacity-achieving codes possessing strong security with nearly linear time encoding and decoding algorithms for any degraded wiretap channel, and the construction of capacity-achieving codes possessing semantic security with linear time encoding and decoding algorithms for erasure wiretap channels. Our analysis relies on a relatively small set of tools. One tool is density evolution, a powerful method for analyzing the behavior of message-passing algorithms on long, random sparse graph codes. Another concept we use extensively is the notion of an expander graph. Expander graphs have powerful properties that allow us to prove adversarial, rather than probabilistic, guarantees for message-passing algorithms. Expander graphs are also useful in the context of the wiretap channel because they provide a method for constructing randomness extractors. Finally, we use several well-known isoperimetric inequalities (Harper's inequality, Azuma's inequality, and the Gaussian Isoperimetric inequality) in our analysis of the duality between lossy source coding and channel coding.by Venkat Bala Chandar.Ph.D
Optimum Power and Rate Allocation for Coded V-BLAST: Average Optimization
An analytical framework for performance analysis and optimization of coded V-BLAST is developed. Average power and/or rate allocations to minimize the outage probability as well as their robustness and dual problems are investigated. Compact, closed-form expressions for the optimum allocations and corresponding system performance are given. The uniform power allocation is shown to be near optimum in the low outage regime in combination with the optimum rate allocation. The average rate allocation provides the largest performance improvement (extra diversity gain), and the average power allocation offers a modest SNR gain limited by the number of transmit antennas but does not increase the diversity gain. The dual problems are shown to have the same solutions as the primal ones. All these allocation strategies are shown to be robust. The reported results also apply to coded multiuser detection and channel equalization systems relying on successive interference cancellation
Gaussian optimizers and other topics in quantum information
The main topic of this thesis is the proof of two fundamental entropic
inequalities for quantum Gaussian channels. Quantum Gaussian channels model the
propagation of electromagnetic waves through optical fibers and free space in
the quantum regime, and provide the mathematical model to determine the maximum
rates achievable by quantum communication devices for communication and quantum
key distribution. The first inequality proven in this thesis is the quantum
Entropy Power Inequality, which provides a lower bound to the output entropy of
the beam-splitter or of the squeezing in terms of the entropies of the two
inputs. The second inequality states that Gaussian states minimize the output
entropy of the one-mode quantum Gaussian attenuator among all the input states
with a given entropy, and was a longstanding conjecture in quantum
communication theory. The generalization of this inequality to the multimode
attenuator would determine its triple trade-off region and its capacity region
for broadcast communication to two receivers. The thesis contains further
results in Gaussian quantum information, quantum statistical mechanics and
relativistic quantum information. The most important of these results concerns
the Eigenstate Thermalization Hypothesis (ETH). The ETH is an assumption in
quantum statistical mechanics stating that the eigenstates of the Hamiltonian
of a system+bath compound look as thermal states if we can access only the
system. We prove that the ETH must hold if the system thermalizes for any
initial product state of the system+bath compound with a well-defined
temperature.Comment: PhD thesis defended on 12 Sep 2016, supervisor: Prof. Vittorio
Giovannett
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