arXiv subjects

Akira Kamatsuka

Publications and source records attributed to Akira Kamatsuka.

14 recordsLinked to original sources

arXiv2026

UMVUE-Type Estimators under Bregman Losses

We study unbiased estimation under Bregman losses and develop an extension of the classical theory of uniformly minimum variance unbiased estimators (UMVUEs). Exploiting bias--variance-type decompositions for Bregman divergences, we consider two natural loss functions, $D_φ(θ,\hatθ)$ and $D_φ(\hatθ,θ)$, and their corresponding notions of unbiasedness. We show that the latter formulation reduces to the classical setting, whereas the former yields a different framework in which unbiasedness is characterized in the dual space induced by $\nablaφ$. For the nontrivial case, we establish analogs of the Rao--Blackwell and Lehmann--Scheff{é} theorems, providing a systematic construction of type-I Bregman UMVUEs.

cs.IT↗

arXiv2026

Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy

This study focuses on conditional entropy frameworks based on the Kolmogorov--Nagumo (KN) mean. First, $(η, ψ)$-KN averaging (\texttt{EPKNAVG}), a KN-mean extension of the $η$-averaging (\texttt{EAVG}) framework for $(η, F)$-entropies, is introduced and proven to be equivalent to \texttt{EAVG} under suitable concavification conditions. Second, motivated by generalized $g$-vulnerability, a new framework is proposed for generalized $g$-conditional entropies. This framework captures conditional entropies beyond the scope of \texttt{EAVG}-type representations. In particular, it is shown that there exists an $α$ and a joint probability distribution $p_{X, Y}$ such that the Augustin--Csisz{\' a}r conditional entropy $H_α^{\mathrm{C}}(X|Y)$ cannot be represented by any $(η,F)$-entropy satisfying \texttt{EAVG}. In contrast, it is represented within the proposed framework. Furthermore, sufficient conditions are derived under which the proposed generalized $g$-conditional entropies satisfy the conditioning reduces entropy property and the data-processing inequality.

cs.IT↗

arXiv2026

A Generalized Information Bottleneck Method: A Decision-Theoretic Perspective

The information bottleneck (IB) method seeks a compressed representation of data that preserves information relevant to a target variable for prediction while discarding irrelevant information from the original data. In its classical formulation, the IB method employs mutual information to evaluate the compression between the original and compressed data and the utility of the representation for the target variable. In this study, we investigate a generalized IB problem, where the evaluation of utility is based on the $\mathcal{H}$-mutual information that satisfies the concave (\texttt{CV}) and averaging (\texttt{AVG}) conditions. This class of information measures admits a statistical decision-theoretic interpretation via its equivalence to the expected value of sample information. Based on this interpretation, we derive an alternating optimization algorithm to assess the tradeoff between compression and utility in the generalized IB problem.

cs.IT↗

arXiv2026

A Generalized Leakage Interpretation of Alpha-Mutual Information

This paper presents a unified interpretation of $α$-mutual information ($α$-MI) in terms of generalized $g$-leakage. Specifically, we present a novel interpretation of $α$-MI within an extended framework for quantitative information flow based on adversarial generalized decision problems. This framework employs the Kolmogorov-Nagumo mean and the $q$-logarithm to characterize adversarial gain. Furthermore, we demonstrate that, within this framework, the parameter $α$ can be interpreted as a measure of the adversary's risk aversion.

cs.IT↗

arXiv2025

Alternating Optimization Approach for Computing $α$-Mutual Information and $α$-Capacity

This study presents alternating optimization (AO) algorithms for computing $α$-mutual information ($α$-MI) and $α$-capacity based on variational characterizations of $α$-MI using a reverse channel. Specifically, we derive several variational characterizations of Sibson, Arimoto, Augustin--Csisz{\' a}r, and Lapidoth--Pfister MI and introduce novel AO algorithms for computing $α$-MI and $α$-capacity; their performances for computing $α$-capacity are also compared. The comparison results show that the AO algorithm based on the Sibson MI's characterization has the fastest convergence speed.

cs.IT↗

arXiv2025

Several Representations of $α$-Mutual Information and Interpretations as Privacy Leakage Measures

In this paper, we present several novel representations of $α$-mutual information ($α$-MI) in terms of R{\' e}nyi divergence and conditional R{\' e}nyi entropy. The representations are based on the variational characterizations of $α$-MI using a reverse channel. Based on these representations, we provide several interpretations of the $α$-MI as privacy leakage measures using generalized mean and gain functions. Further, as byproducts of the representations, we propose novel conditional R{\' e}nyi entropies that satisfy the property that conditioning reduces entropy and data-processing inequality.

cs.IT↗

arXiv2024

A Variational Characterization of $H$-Mutual Information and its Application to Computing $H$-Capacity

$H$-mutual information ($H$-MI) is a wide class of information leakage measures, where $H=(η, F)$ is a pair of monotonically increasing function $η$ and a concave function $F$, which is a generalization of Shannon entropy. $H$-MI is defined as the difference between the generalized entropy $H$ and its conditional version, including Shannon mutual information (MI), Arimoto MI of order $α$, $g$-leakage, and expected value of sample information. This study presents a variational characterization of $H$-MI via statistical decision theory. Based on the characterization, we propose an alternating optimization algorithm for computing $H$-capacity.

cs.IT↗

arXiv2024

New Algorithms for Computing Sibson Capacity and Arimoto Capacity

The Sibson and Arimoto capacity, which are based on the Sibson and Arimoto mutual information (MI) of order α, respectively, are well-known generalizations of the channel capacity C. In this study, we derive novel alternating optimization algorithms for computing these capacities by providing new variational characterizations of the Sibson and Arimoto MI. Moreover, we prove that all iterative algorithms for computing these capacities are equivalent under appropriate conditions imposed on their initial distributions.

cs.IT↗

arXiv2024

A New Algorithm for Computing $α$-Capacity

The problem of computing $α$-capacity for $α>1$ is equivalent to that of computing the correct decoding exponent. Various algorithms for computing them have been proposed, such as Arimoto and Jitsumatsu--Oohama algorithm. In this study, we propose a novel alternating optimization algorithm for computing the $α$-capacity for $α>1$ based on a variational characterization of the Augustin--Csisz{á}r mutual information. A comparison of the convergence performance of these algorithms is demonstrated through numerical examples.

cs.IT↗

arXiv2022

On Hypothesis Testing via a Tunable Loss

We consider a problem of simple hypothesis testing using a randomized test via a tunable loss function proposed by Liao \textit{et al}. In this problem, we derive results that correspond to the Neyman--Pearson lemma, the Chernoff--Stein lemma, and the Chernoff-information in the classical hypothesis testing problem. Specifically, we prove that the optimal error exponent of our problem in the Neyman--Pearson's setting is consistent with the classical result. Moreover, we provide lower bounds of the optimal Bayesian error exponent.

cs.IT↗

arXiv2022

An Algorithm for Computing the Stratonovich's Value of Information

We propose an algorithm for computing Stratonovich's value of information (VoI) that can be regarded as an analogue of the distortion-rate function. We construct an alternating optimization algorithm for VoI under a general information leakage constraint and derive a convergence condition. Furthermore, we discuss algorithms for computing VoI under specific information leakage constraints, such as Shannon's mutual information (MI), $f$-leakage, Arimoto's MI, Sibson's MI, and Csiszar's MI.

cs.IT↗

arXiv2022

A Generalization of the Stratonovich's Value of Information and Application to Privacy-Utility Trade-off

The Stratonovich's value of information (VoI) is quantity that measure how much inferential gain is obtained from a perturbed sample under information leakage constraint. In this paper, we introduce a generalized VoI for a general loss function and general information leakage. Then we derive an upper bound of the generalized VoI. Moreover, for a classical loss function, we provide a achievable condition of the upper bound which is weaker than that of in previous studies. Since VoI can be viewed as a formulation of a privacy-utility trade-off (PUT) problem, we provide an interpretation of the achievable condition in the PUT context.

cs.IT↗

arXiv2022

Probability Distribution on Rooted Trees

The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical expressive capability causes a problem in tree selection to avoid overfitting. One unified approach to solve this is a Bayesian approach, on which the rooted tree is regarded as a random variable and a direct loss function can be assumed on the selected model or the predicted value for a new data point. However, all the previous studies on this approach are based on the probability distribution on full trees, to the best of our knowledge. In this paper, we propose a generalized probability distribution for any rooted trees in which only the maximum number of child nodes and the maximum depth are fixed. Furthermore, we derive recursive methods to evaluate the characteristics of the probability distribution without any approximations.

cs.LG↗

arXiv2022

Probability Distribution on Full Rooted Trees

The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting becomes problematic. A method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on the Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.

stat.ML↗