arXiv subjects

Ken-ichi Iwata

Publications and source records attributed to Ken-ichi Iwata.

15 recordsLinked to original sources

arXiv2026

Asymmetric Encoding-Decoding Schemes for Lossless Data Compression

This paper proposes a new lossless data compression coding scheme named an asymmetric encoding-decoding scheme (AEDS), which can be considered as a generalization of tANS (tabled variant of asymmetric numeral systems). In the AEDS, a data sequence $\mathbf{s}=s_1s_2\cdots s_n$ is encoded in backward order $s_t, t=n, \cdots, 2,1$, while $\mathbf{s}$ is decoded in forward order $s_t, t=1, 2, \cdots, n$ in the same way as the tANS. But, the code class of the AEDS is much broader than that of the tANS. We show for i.i.d.~sources that an AEDS with 2 states (resp.~5 states) can attain a shorter average code length than the Huffman code if a child of the root in the Huffman code tree has a probability weight larger than 0.61803 (resp.~0.56984). Furthermore, we derive several upper bounds on the average code length of the AEDS, which also hold for the tANS, and we show that the average code length of the optimal AEDS and tANS with $N$ states converges to the source entropy with speed $O(1/N)$ as $N$ increases.

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arXiv2024

Reduction of Sufficient Number of Code Tables of $k$-Bit Delay Decodable Codes

A $k$-bit delay decodable code-tuple is a lossless source code that can achieve a smaller average codeword length than Huffman codes by using a finite number of code tables and allowing at most $k$-bit delay for decoding. It is known that there exists a $k$-bit delay decodable code-tuple with at most $2^{(2^k)}$ code tables that attains the optimal average codeword length among all the $k$-bit delay decodable code-tuples for any given i.i.d. source distribution. Namely, it suffices to consider only the code-tuples with at most $2^{(2^k)}$ code tables to accomplish optimality. In this paper, we propose a method to dramatically reduce the number of code tables to be considered in the theoretical analysis, code construction, and coding process.

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arXiv2024

Encoding and Decoding Algorithms of ANS Variants and Evaluation of Their Average Code Lengths

Asymmetric Numeral Systems (ANS) proposed by Jarek Duda are high-performance distortionless data compression schemes that can achieve almost the same compression performance as arithmetic codes with less arithmetic operations than arithmetic coding. The ANS is widely used in various practical systems like Facebook, Apple, Google, Dropbox, Microsoft, and Pixar, due to their high performance, but many researchers still lack much knowledge about the ANS. This paper thoroughly explains the encoding and decoding algorithms of the ANS, and theoretically analyzes the average code length achievable by the ANS.

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arXiv2023

The Optimality of AIFV Codes in the Class of $2$-bit Delay Decodable Codes

AIFV (almost instantaneous fixed-to-variable length) codes are noiseless source codes that can attain a shorter average codeword length than Huffman codes by allowing a time-variant encoder with two code tables and a decoding delay of at most 2 bits. First, we consider a general class of noiseless source codes, called k-bit delay decodable codes, in which one allows a finite number of code tables and a decoding delay of at most k bits for k >= 0. Then we prove that AIFV codes achieve the optimal average codeword length in the 2-bit delay decodable codes class.

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arXiv2023

Properties of k-bit Delay Decodable Codes

The class of k-bit delay decodable codes, source codes allowing decoding delay of at most k bits for k >= 0, can attain a shorter average codeword length than Huffman codes. This paper discusses the general properties of the class of k-bit delay decodable codes with a finite number of code tables and proves two theorems which enable us to limit the scope of code-tuples to be considered when discussing optimal k-bit delay decodable code-tuples.

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arXiv2022

Optimality of Huffman Code in the Class of 1-bit Delay Decodable Codes

For a given independent and identically distributed (i.i.d.) source, Huffman code achieves the optimal average codeword length in the class of instantaneous code with a single code table. However, it is known that there exist time-variant encoders, which achieve a shorter average codeword length than the Huffman code, using multiple code tables and allowing at most k-bit decoding delay for k = 2, 3, 4, . . .. On the other hand, it is not known whether there exists a 1-bit delay decodable code, which achieves a shorter average length than the Huffman code. This paper proves that for a given i.i.d. source, a Huffman code achieves the optimal average codeword length in the class of 1-bit delay decodable codes with a finite number of code tables.

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arXiv2020

Modular Arithmetic Erasure Channels and Their Multilevel Channel Polarization

This study proposes \emph{modular arithmetic erasure channels} (MAECs), a novel class of erasure-like channels with an input alphabet that need not be binary. This class contains the binary erasure channel (BEC) and some other known erasure-like channels as special cases. For MAECs, we provide recursive formulas of Arıkan-like polar transform to simulate channel polarization. In other words, we show that the synthetic channels of MAECs are equivalent to other MAECs. This is a generalization of well-known recursive formulas of the polar transform for BECs. Using our recursive formulas, we also show that a recursive application of the polar transform for MAECs results in \emph{multilevel channel polarization,} which is an asymptotic phenomenon that is characteristic of non-binary polar codes. Specifically, we establish a method to calculate the limiting proportions of the partially noiseless and noisy channels that are generated as a result of multilevel channel polarization for MAECs. In the particular case of MAECs, this calculation method solves an open problem posed by Nasser (2017) in the study of non-binary polar codes.

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arXiv2019

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 Arıkan-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.

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arXiv2018

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.

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arXiv2017

Proceedings of Workshop AEW10: Concepts in Information Theory and Communications

The 10th Asia-Europe workshop in "Concepts in Information Theory and Communications" AEW10 was held in Boppard, Germany on June 21-23, 2017. It is based on a longstanding cooperation between Asian and European scientists. The first workshop was held in Eindhoven, the Netherlands in 1989. The idea of the workshop is threefold: 1) to improve the communication between the scientist in the different parts of the world; 2) to exchange knowledge and ideas; and 3) to pay a tribute to a well respected and special scientist.

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arXiv2017

Sharp Bounds on Arimoto's Conditional Rényi Entropies Between Two Distinct Orders

This study examines sharp bounds on Arimoto's conditional Rényi entropy of order $β$ with a fixed another one of distinct order $α\neq β$. Arimoto inspired the relation between the Rényi entropy and the $\ell_{r}$-norm of probability distributions, and he introduced a conditional version of the Rényi entropy. From this perspective, we analyze the $\ell_{r}$-norms of particular distributions. As results, we identify specific probability distributions whose achieve our sharp bounds on the conditional Rényi entropy. The sharp bounds derived in this study can be applicable to other information measures, e.g., the minimum average probability of error, the Bhattacharyya parameter, Gallager's reliability function $E_{0}$, and Sibson's $α$-mutual information, whose are strictly monotone functions of the conditional Rényi entropy.

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arXiv2016

Sharp Bounds Between Two Rényi Entropies of Distinct Positive Orders

Many axiomatic definitions of entropy, such as the Rényi entropy, of a random variable are closely related to the $\ell_α$-norm of its probability distribution. This study considers probability distributions on finite sets, and examines the sharp bounds of the $\ell_β$-norm with a fixed $\ell_α$-norm, $α\neq β$, for $n$-dimensional probability vectors with an integer $n \ge 2$. From the results, we derive the sharp bounds of the Rényi entropy of positive order $β$ with a fixed Rényi entropy of another positive order $α$. As applications, we investigate sharp bounds of Ariomoto's mutual information of order $α$ and Gallager's random coding exponents for uniformly focusing channels under the uniform input distribution.

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arXiv2016

A Generalized Erasure Channel in the Sense of Polarization for Binary Erasure Channels

The polar transformation of a binary erasure channel (BEC) can be exactly approximated by other BECs. Arıkan proposed that polar codes for a BEC can be efficiently constructed by using its useful property. This study proposes a new class of arbitrary input generalized erasure channels, which can be exactly approximated the polar transformation by other same channel models, as with the BEC. One of the main results is the recursive formulas of the polar transformation of the proposed channel. In the study, we evaluate the polar transformation by using the $α$-mutual information. Particularly, when the input alphabet size is a prime power, we examines the following: (i) inequalities for the average of the $α$-mutual information of the proposed channel after the one-step polar transformation, and (ii) the exact proportion of polarizations of the $α$-mutual information of proposed channels in infinite number of polar transformations.

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arXiv2016

Relations Between Conditional Shannon Entropy and Expectation of $\ell_α$-Norm

The paper examines relationships between the conditional Shannon entropy and the expectation of $\ell_α$-norm for joint probability distributions. More precisely, we investigate the tight bounds of the expectation of $\ell_α$-norm with a fixed conditional Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the conditional Shannon entropy and several information measures which are determined by the expectation of $\ell_α$-norm, e.g., the conditional Rényi entropy and the conditional $R$-norm information. Moreover, we apply these results to discrete memoryless channels under a uniform input distribution. Then, we show the tight bounds of Gallager's $E_{0}$ functions with a fixed mutual information under a uniform input distribution.

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arXiv2016

Extremal Relations Between Shannon Entropy and $\ell_α$-Norm

The paper examines relationships between the Shannon entropy and the $\ell_α$-norm for $n$-ary probability vectors, $n \ge 2$. More precisely, we investigate the tight bounds of the $\ell_α$-norm with a fixed Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds between the Shannon entropy and several information measures which are determined by the $\ell_α$-norm, e.g., Rényi entropy, Tsallis entropy, the $R$-norm information, and some diversity indices. Moreover, we apply these results to uniformly focusing channels. Then, we show the tight bounds of Gallager's $E_{0}$ functions with a fixed mutual information under a uniform input distribution.

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