arXiv subjects

Yoshinori Takei

Publications and source records attributed to Yoshinori Takei.

7 recordsLinked to original sources

arXiv2025

A Simple Recursive Relation Characterizes a Tree Associated to Generalized Farey Sequences

This paper proves that two differently defined rooted binary trees are isomorphic. The first tree is one associated to a version of Farey sequences where the vertices correspond to the open intervals formed by two successive terms in the sequence. The other tree has the vertices consisting of pairs of positive integers whose adjacency is defined by a simple recursive relation. These trees appeared in a study of a generalization of a class of the permutations defined by Sós and the bijection between it and the set of the Farey intervals due to Surányi.

math.CO↗

arXiv2025

A Sequential Computation Algorithm for the Center of the Smallest Enclosing Ball

In this paper, we consider the problem of finding the center $Q^\ast$ of the SEB (smallest enclosing ball) for $n$ points in $d$-dimensional Euclidean space. One application of the SEB is SVDD (support vector data description) in support vector machines. Our objective is to develop a sequential computation algorithm for determining the barycentric coordinate of $Q^\ast$. To achieve it, we apply the concept of the Arimoto-Blahut algorithm, which is a sequential computation algorithm used to compute the channel capacity. We first consider the case in which an equidistant point $\widetilde{Q}$ from the $n$ points exists, and construct a recurrence formula that converges to the barycentric coordinate $\widetilde{\bmλ}$ of $\widetilde{Q}$. When $\widetilde{Q}$ lies within the convex hull of the $n$ points, $\widetilde{Q}$ coincides with $Q^\ast$, hence in this case, the recurrence formula converges to the barycentric coordinate $\bmλ^\ast$ of $Q^\ast$. The resulting recurrence formula is very simple because it uses only the coordinates of the $n$ points. The computational complexity, with an approximation error of $ε$ to the exact solution $\widetilde{\bmλ}$, is $O(κn^2\log(1/ε))$, where $κ$ is a value determined by the $n$ points. Furthermore, we modify the algorithm so that it can also be applied in cases where $\widetilde{Q}$ does not exist, and evaluate the convergence performance numerically. We compare the proposed algorithm with conventional algorithms in terms of run time and computational accuracy through several examples. The proposed algorithm has some advantages and some disadvantages compared to the conventional algorithms, but overall, since the proposed algorithm can be computed using a very simple formula, it is considered sufficiently practical.

cs.IT↗

arXiv2024

A congruential recurrence characterizes the inverses of Sós permutations

In a proof of the three gaps theorem, a class of permutations known as the Sós permutations was introduced. It is known that a Sós permutation, as a sequence, satisfies a certain recurrence (Sós's recurrence), however, whether the converse holds remains unknown. On the other hand, the inverses of Sós permutations have been studied also. It has been reported that such a permutation satisfies a congruential recurrence as a sequence. The converse problem of this fact, i.e., whether a permutation satisfying the congruential recurrence is the inverse of a Sós permutation, is also unsolved, except for a finite number of the degrees of the permutations. This paper relates the set of permutations satisfying the congruential recurrence to other sets of permutations and gives upper bounds for their cardinalities. The upper bounds are in fact tight. In particular, the set of permutations satisfying the congruential recurrence has the same cardinality as that of Sós permutations, giving the affirmative answer to the above unsolved problem as a corollary. As another corollary, it is shown that all permutation, which is regarded as a congruential quasi-progression of diameter 1 in that the set formed by the first order differences modulo the degree of the permutation is a singleton or a set of two successive integers, is the inverse of a Sós permutation with an elementary operation called as shift applied. As an application of these facts, we present a procedure that lifts the set of the inverses of the Sós permutations of a given degree to the set of the same kind with the degree increased by one, without referring to the underlying parameters defining the Sós permutations or the Farey sequence associated to them.

math.CO↗

arXiv2022

On a Proof of the Convergence Speed of a Second-order Recurrence Formula in the Arimoto-Blahut Algorithm

In [8] (Nakagawa, et.al., IEEE Trans. IT, 2021), we investigated the convergence speed of the Arimoto-Blahut algorithm. In [8], the convergence of the order $O(1/N)$ was analyzed by focusing on the second-order nonlinear recurrence formula consisting of the first- and second-order terms of the Taylor expansion of the defining function of the Arimoto-Blahut algorithm. However, in [8], an infinite number of inequalities were assumed as a "conjecture," and proofs were given based on the conjecture. In this paper, we report a proof of the convergence of the order $O(1/N)$ for a class of channel matrices without assuming the conjecture. The correctness of the proof will be confirmed by several numerical examples.

cs.IT↗

arXiv2020

Analysis of the Convergence Speed of the Arimoto-Blahut Algorithm by the Second Order Recurrence Formula

In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices the convergence is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence is new in this paper. Based on the analysis, we will compare the convergence speed of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.

cs.IT↗

arXiv2018

Analysis for the Slow Convergence in Arimoto Algorithm

In this paper, we investigate the convergence speed of the Arimoto algorithm. By analyzing the Taylor expansion of the defining function of the Arimoto algorithm, we will clarify the conditions for the exponential or $1/N$ order convergence and calculate the convergence speed. We show that the convergence speed of the $1/N$ order is evaluated by the derivatives of the Kullback-Leibler divergence with respect to the input probabilities. The analysis for the convergence of the $1/N$ order is new in this paper. Based on the analysis, we will compare the convergence speed of the Arimoto algorithm with the theoretical values obtained in our theorems for several channel matrices.

cs.IT↗

arXiv2018

On the Hardness of Deriving the Arithmetic Mean Component Competitive Ratio

For the multi-objective time series search problem, Hasegawa and Itoh [Theoretical Computer Science, Vo.718, pp.58-66, 2018] presented the best possible online algorithm balanced price policy (BPP for short) for any monotone function $f: R^k \to R$. Specifically, the competitive ratio with respect to the monotone function $f(c_{1},\ldots,c_{k})=(c_{1}+\cdots+c_{k})/k$ is referred to as the arithmetic mean component competitive ratio. Hasegawa and Itoh derived the closed formula of the arithmetic mean component competitive ratio for $k=2$, but it has not been known for any integer $k \geq 3$. In this paper, we show that it is NP-hard to derive closed formulas of the arithmetic mean component competitive ratio for general integer $k\geq 2$. On the the hand, we derive closed formulas of the arithmetic mean component competitive ratio for $k=3$ and $k=4$.

cs.DS↗