90 research outputs found
Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities
This monograph presents a unified treatment of single- and multi-user
problems in Shannon's information theory where we depart from the requirement
that the error probability decays asymptotically in the blocklength. Instead,
the error probabilities for various problems are bounded above by a
non-vanishing constant and the spotlight is shone on achievable coding rates as
functions of the growing blocklengths. This represents the study of asymptotic
estimates with non-vanishing error probabilities.
In Part I, after reviewing the fundamentals of information theory, we discuss
Strassen's seminal result for binary hypothesis testing where the type-I error
probability is non-vanishing and the rate of decay of the type-II error
probability with growing number of independent observations is characterized.
In Part II, we use this basic hypothesis testing result to develop second- and
sometimes, even third-order asymptotic expansions for point-to-point
communication. Finally in Part III, we consider network information theory
problems for which the second-order asymptotics are known. These problems
include some classes of channels with random state, the multiple-encoder
distributed lossless source coding (Slepian-Wolf) problem and special cases of
the Gaussian interference and multiple-access channels. Finally, we discuss
avenues for further research.Comment: Further comments welcom
Nonasymptotic noisy lossy source coding
This paper shows new general nonasymptotic achievability and converse bounds
and performs their dispersion analysis for the lossy compression problem in
which the compressor observes the source through a noisy channel. While this
problem is asymptotically equivalent to a noiseless lossy source coding problem
with a modified distortion function, nonasymptotically there is a noticeable
gap in how fast their minimum achievable coding rates approach the common
rate-distortion function, as evidenced both by the refined asymptotic analysis
(dispersion) and the numerical results. The size of the gap between the
dispersions of the noisy problem and the asymptotically equivalent noiseless
problem depends on the stochastic variability of the channel through which the
compressor observes the source.Comment: IEEE Transactions on Information Theory, 201
Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels
We propose a quantum soft-covering problem for a given general quantum
channel and one of its output states, which consists in finding the minimum
rank of an input state needed to approximate the given channel output. We then
prove a one-shot quantum covering lemma in terms of smooth min-entropies by
leveraging decoupling techniques from quantum Shannon theory. This covering
result is shown to be equivalent to a coding theorem for rate distortion under
a posterior (reverse) channel distortion criterion [Atif, Sohail, Pradhan,
arXiv:2302.00625]. Both one-shot results directly yield corollaries about the
i.i.d. asymptotics, in terms of the coherent information of the channel.
The power of our quantum covering lemma is demonstrated by two additional
applications: first, we formulate a quantum channel resolvability problem, and
provide one-shot as well as asymptotic upper and lower bounds. Secondly, we
provide new upper bounds on the unrestricted and simultaneous identification
capacities of quantum channels, in particular separating for the first time the
simultaneous identification capacity from the unrestricted one, proving a
long-standing conjecture of the last author.Comment: 29 pages, 3 figures; v2 fixes an error in Definition 6.1 and various
typos and minor issues throughou
Variable-length compression allowing errors
This paper studies the fundamental limits of the minimum average length of
lossless and lossy variable-length compression, allowing a nonzero error
probability , for lossless compression. We give non-asymptotic bounds
on the minimum average length in terms of Erokhin's rate-distortion function
and we use those bounds to obtain a Gaussian approximation on the speed of
approach to the limit which is quite accurate for all but small blocklengths:
where is the functional
inverse of the standard Gaussian complementary cdf, and is the
source dispersion. A nonzero error probability thus not only reduces the
asymptotically achievable rate by a factor of , but this
asymptotic limit is approached from below, i.e. larger source dispersions and
shorter blocklengths are beneficial. Variable-length lossy compression under an
excess distortion constraint is shown to exhibit similar properties
Successive Refinement of Shannon Cipher System Under Maximal Leakage
We study the successive refinement setting of Shannon cipher system (SCS)
under the maximal leakage constraint for discrete memoryless sources under
bounded distortion measures. Specifically, we generalize the threat model for
the point-to-point rate-distortion setting of Issa, Wagner and Kamath (T-IT
2020) to the multiterminal successive refinement setting. Under mild conditions
that correspond to partial secrecy, we characterize the asymptotically optimal
normalized maximal leakage region for both the joint excess-distortion
probability (JEP) and the expected distortion reliability constraints. Under
JEP, in the achievability part, we propose a type-based coding scheme, analyze
the reliability guarantee for JEP and bound the leakage of the information
source through compressed versions. In the converse part, by analyzing a
guessing scheme of the eavesdropper, we prove the optimality of our
achievability result. Under expected distortion, the achievability part is
established similarly to the JEP counterpart. The converse proof proceeds by
generalizing the corresponding results for the rate-distortion setting of SCS
by Schieler and Cuff (T-IT 2014) to the successive refinement setting. Somewhat
surprisingly, the normalized maximal leakage regions under both JEP and
expected distortion constraints are identical under certain conditions,
although JEP appears to be a stronger reliability constraint
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