arXiv subjects

Hiroki Hashiguchi

Publications and source records attributed to Hiroki Hashiguchi.

13 recordsLinked to original sources

arXiv2025

Evaluating Singular Value Thresholds for DNN Weight Matrices based on Random Matrix Theory

This study evaluates thresholds for removing singular values from singular value decomposition-based low-rank approximations of deep neural network weight matrices. Each weight matrix is modeled as the sum of signal and noise matrices. The low-rank approximation is obtained by removing noise-related singular values using a threshold based on random matrix theory. To assess the adequacy of this threshold, we propose an evaluation metric based on the cosine similarity between the singular vectors of the signal and original weight matrices. The proposed metric is used in numerical experiments to compare two threshold estimation methods.

stat.ML↗

arXiv2025

Exact Distribution of the Noncentral Complex Roy's Largest Root Statistic via Pieri's Formula

In this study, we derive the exact distribution and moment of the noncentral complex Roy's largest root statistic, expressed as a product of complex zonal polynomials. We show that the linearization coefficients arising from the product of complex zonal polynomials in the distribution of Roy's test under a specific alternative hypothesis can be explicitly computed using Pieri's formula, a well-known result in combinatorics. These results were then applied to compute the power of tests in the complex multivariate analysis of variance (MANOVA).

math.ST↗

arXiv2023

Chi-square approximation for the distribution of individual eigenvalues of a singular Wishart matrix

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hyper-geometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations

math.ST↗

arXiv2022

Numerical computation for the exact distribution of Roy's largest root statistic under linear alternative

This paper discusses the computation of exact powers for Roy's test in multivariate analysis of variance~(MANOVA). We derive an exact expression for the largest eigenvalue of a singular noncentral Beta matrix in terms of the product of zonal polynomials. The numerical computation for that distribution is conducted by an algorithm that expands the product of zonal polynomials as a linear combination of zonal polynomials. Furthermore, we provide an exact distribution of the largest eigenvalue in a form that is convenient for numerical calculations under the linear alternative.

math.ST↗

arXiv2021

Algorithm for the product of Jack polynomials and its application to the sphericity test

In this study, we derive the density and distribution function of a ratio of the largest and smallest eigenvalues of a singular beta-Wishart matrix for the sphericity test. These functions can be expressed in terms of the product of Jack polynomials. We propose an algorithm that expands the product of Jack polynomials by a linear combination of Jack polynomials. Numerical computation for the derived distributions is performed using the algorithm.

math.ST↗

arXiv2021

Generalized heterogeneous hypergeometric functions and the distribution of the largest eigenvalue of an elliptical Wishart matrix

In this study, we derive the exact distributions of eigenvalues of a singular Wishart matrix under an elliptical model. We define generalized heterogeneous hypergeometric functions with two matrix arguments and provide convergence conditions for these functions. The joint density of eigenvalues and the distribution function of the largest eigenvalue for a singular elliptical Wishart matrix are represented by these functions. Numerical computations for the distribution of the largest eigenvalue were conducted under the matrix-variate $t$ and Kotz-type models.

math.ST↗

arXiv2021

Approximation to probability density functions in sampling distributions based on Fourier cosine series

We derive a simple and precise approximation to probability density functions in sampling distributions based on the Fourier cosine series. After clarifying the required conditions, we illustrate the approximation on two examples: the distribution of the sum of uniformly distributed random variables, and the distribution of sample skewness drawn from a normal population. The probability density function of the first example can be explicitly expressed, but that of the second example has no explicit expression.

math.ST↗

arXiv2020

Illuminating the electronic properties of WS$_2$ polytypism with electron microscopy

Tailoring the specific stacking sequences (polytypes) of layered materials represents a powerful strategy to identify and design novel physical properties. While nanostructures built upon transition-metal dichalcogenides (TMDs) with either the 2H or 3R crystalline phases have been routinely studied, our knowledge of those based on mixed 2H/3R polytypes is far more limited. Here we report on the characterization of mixed 2H/3R free-standing WS$_2$ nanostructures displaying a flower-like configuration by means of advanced transmission electron microscopy. We correlate their rich variety of shape-morphology combinations with relevant local electronic properties such as their edge, surface, and bulk plasmons. Electron energy-loss spectroscopy combined with machine learning reveals that the 2H/3R polytype displays an indirect band gap with $E_{\rm BG}=1.6^{+0.3}_{-0.2}~{\rm eV}$. Further, we identify the presence of energy-gain peaks in the EEL spectra characterized by a gain-to-loss ratio $I_G/I_L>1$. Such property could be exploited to develop novel cooling strategies for atomically thin TMD nanostructures and devices built upon them. Our results represent a stepping stone towards an improved understanding of TMDs nanostructures based on mixed crystalline phases.

cond-mat.mtrl-sci↗

arXiv2020

Expressing the largest eigenvalue of a singular beta F-matrix with heterogeneous hypergeometric functions

In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of a nonsingular beta F -matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.

math.ST↗

arXiv2019

Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix

This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the eigenvalues of singular beta-Wishart matrices. The joint density function of the eigenvalues and the distribution of the largest eigenvalue can be expressed in terms of certain heterogeneous hypergeometric functions. Exact computation of the distribution of the largest eigenvalue is conducted here for a real case.

math.ST↗

arXiv2016

Distribution of ratio of two Wishart matrices and evaluation of cumulative probability by holonomic gradient method

We study the distribution of the ratio of two central Wishart matrices with different covariance matrices. We first derive the density function of a particular matrix form of the ratio and show that its cumulative distribution function can be expressed in terms of the hypergeometric function 2F1 of a matrix argument. Then we apply the holonomic gradient method for numerical evaluation of the hypergeometric function. This approach enables us to compute the power function of Roy's maximum root test for testing the equality of two covariance matrices.

math.ST↗

arXiv2013

Holonomic gradient method for the distribution function of the largest root of a Wishart matrix

We apply the holonomic gradient method introduced by Nakayama et al.(2011) to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function 1F1 of a matrix argument. Numerical evaluation of the hypergeometric function has been one of the longstanding problems in multivariate distribution theory. The holonomic gradient method offers a totally new approach, which is complementary to the infinite series expansion around the origin in terms of zonal polynomials. It allows us to move away from the origin by the use of partial differential equations satisfied by the hypergeometric function. From numerical viewpoint we show that the method works well up to dimension 10. From theoretical viewpoint the method offers many challenging problems both to statistics and D-module theory.

math.ST↗

arXiv2012

A measure of skewness for testing departures from normality

We propose a new skewness test statistic for normality based on the Pearson measure of skewness. We obtain asymptotic first four moments of the null distribution for this statistic by using a computer algebra system and its normalizing transformation based on the Johnson $S_{U}$ system. Finally the performance of the proposed statistic is shown by comparing the powers of several skewness test statistics against some alternative hypotheses.

stat.CO↗