312 research outputs found
Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory
Fano's inequality is one of the most elementary, ubiquitous, and important
tools in information theory. Using majorization theory, Fano's inequality is
generalized to a broad class of information measures, which contains those of
Shannon and R\'{e}nyi. When specialized to these measures, it recovers and
generalizes the classical inequalities. Key to the derivation is the
construction of an appropriate conditional distribution inducing a desired
marginal distribution on a countably infinite alphabet. The construction is
based on the infinite-dimensional version of Birkhoff's theorem proven by
R\'{e}v\'{e}sz [Acta Math. Hungar. 1962, 3, 188{\textendash}198], and the
constraint of maintaining a desired marginal distribution is similar to
coupling in probability theory. Using our Fano-type inequalities for Shannon's
and R\'{e}nyi's information measures, we also investigate the asymptotic
behavior of the sequence of Shannon's and R\'{e}nyi's equivocations when the
error probabilities vanish. This asymptotic behavior provides a novel
characterization of the asymptotic equipartition property (AEP) via Fano's
inequality.Comment: 44 pages, 3 figure
Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions
Polar transforms are central operations in the study of polar codes. This
paper examines polar transforms for non-stationary memoryless sources on
possibly infinite source alphabets. This is the first attempt of source
polarization analysis over infinite alphabets. The source alphabet is defined
to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar
transform based on the group. Defining erasure distributions based on the
normal subgroup structure, we give recursive formulas of the polar transform
for our proposed erasure distributions. As a result, the recursive formulas
lead to concrete examples of multilevel source polarization with countably
infinite levels when the group is locally cyclic. We derive this result via
elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019
IEEE International Symposium on Information Theory (ISIT2019
Asymptotic Distribution of Multilevel Channel Polarization for a Certain Class of Erasure Channels
This study examines multilevel channel polarization for a certain class of
erasure channels that the input alphabet size is an arbitrary composite number.
We derive limiting proportions of partially noiseless channels for such a
class. The results of this study are proved by an argument of convergent
sequences, inspired by Alsan and Telatar's simple proof of polarization, and
without martingale convergence theorems for polarization process.Comment: 31 pages; 1 figure; 1 table; a short version of this paper has been
submitted to the 2018 IEEE International Symposium on Information Theory
(ISIT2018
Clifford's theorem for bricks
Let be a finite group, a normal subgroup of , and a field of
characteristic . In this paper, we formulate the brick version of
Clifford's theorem under suitable assumptions and prove it by using the theory
of wide subcategories. As an application of our theorem, we consider the
restrictions of semibricks and two-term simple-minded collections under the
assumption that the index of the normal subgroup in is a -power
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