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Nobuki Takayama

Publications and source records attributed to Nobuki Takayama.

At least 19 recordsLinked to original sources
arXiv2026

Comparison of Numerical Solvers for Differential Equations for Holonomic Gradient Method in Statistics

Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation (ODE) with respect to a curve in the parameter space. We can evaluate the integral by solving the linear ODE numerically. This approach to evaluate numerically definite integrals is called the holonomic gradient method (HGM) and it is useful to evaluate several normalizing constants in statistics. We will discuss and compare methods to solve linear ODE's to evaluate normalizing constants.

math.NA↗
arXiv2025

Algorithm for direct sampling from conditional distributions of toric models

We show that contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from the conditional distribution of toric models in statistics. The algorithm is based on a Markov chain on a lattice generated by a matrix $A$. A correspondence between decomposable graphical models and $A$-hypergeometric systems is discussed. We give a sum formula of special values of $A$-hypergeometric polynomials. Some examples with implementations are presented.

math.ST↗
arXiv2025

Comprehensive Restriction Algorithm for Hypergeometric Systems

An algorithm computing the restriction of a holonomic D-module to a linear subspace was given by T.Oaku in 1997. We consider a problem of computing the restriction for a given holonomic D-module with parameters. We will give a partial answer to the problem for general holonomic D-modules and an answer to hypergeometric holonomic D-modules.

math.AC↗
arXiv2025

Differential Space of Feynman Integrals: Annihilators and $\mathcal{D}$-module

We present a novel algorithm for constructing differential operators with respect to external variables that annihilate Feynman-like integrals and give rise to the associated $\mathcal{D}$-modules, based on Griffiths-Dwork reduction. By leveraging the Macaulay matrix method, we derive corresponding relations among partial differential operators, including systems of Pfaffian equations and Picard-Fuchs operators. Our computational approach is applicable to twisted period integrals in projective coordinates, and we showcase its application to Feynman graphs and Witten diagrams. The method yields annihilators and their algebraic relations for generic regulator values, explicitly avoiding contributions from surface terms. In the cases examined, we observe that the holonomic rank of the $\mathcal{D}$-modules coincides with the dimension of the corresponding de Rham co-homology groups, indicating an equivalence relation between them, which we propose as a conjecture.

hep-th↗
arXiv2024

An Application of the Holonomic Gradient Method to the Neural Tangent Kernel

A holonomic system of linear partial differential equations is, roughly speaking, a system whose solution space is finite dimensional. A distribution that is a solution of a holonomic system is called a holonomic distribution. We give methods to numerically evaluate dual activations of holonomic activator distributions for neural tangent kernels. These methods are based on computer algebra algorithms for rings of differential operators.

cs.LG↗
arXiv2023

Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers

We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for ${\cal A}$-hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

hep-th↗
arXiv2023

Restrictions of Pfaffian Systems for Feynman Integrals

This work studies limits of Pfaffian systems, a class of first-order PDEs appearing in the Feynman integral calculus. Such limits appear naturally in the context of scattering amplitudes when there is a separation of scale in a given set of kinematic variables. We model these limits, which are often singular, via restrictions of D-modules. We thereby develop two different restriction algorithms: one based on gauge transformations, and another relying on the Macaulay matrix. These algorithms output Pfaffian systems containing fewer variables and of smaller rank. We show that it is also possible to retain logarithmic corrections in the limiting variable. The algorithms are showcased in many examples involving Feynman integrals and hypergeometric functions coming from GKZ systems.

hep-th↗
arXiv2021

Fast evaluation of some p-adic transcendental functions

We design algorithms for computing values of many p-adic elementary and special functions, including logarithms, exponentials, polylogarithms, and hypergeometric functions. All our algorithms feature a quasi-linear complexity with respect to the target precision and most of them are based on an adaptation to the-adic setting of the binary splitting and bit-burst strategies.

cs.SC↗
arXiv2020

Computation of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Non-central Wishart Matrix

We give an approximate formula for the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for the general dimension. The formula is expressed in terms of a definite integral with parameters. We derive a differential equation satisfied by the integral for the $2 \times 2$ matrix case and perform a numerical analysis of it.

math.ST↗
arXiv2019

Holonomic Gradient Method for Two Way Contingency Tables

The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. We apply the holonomic gradient method in the case of the conditional Poisson or multinomial distribution on two way contingency tables. We utilize the modular method in computer algebra for an efficient and exact evaluation, and we discuss on complexities of these algorithms and their implementation. We also discuss on a theoretical aspect of the distribution from the viewpoint of the conditional maximum likelihood estimation.

math.CA↗
arXiv2018

Holonomic Gradient Method-Based CDF Evaluation for the Largest Eigenvalue of a Complex Noncentral Wishart Matrix

The outage probability of maximal-ratio combining (MRC) for a multiple-input multiple-output (MIMO) wireless communications system under Rician fading is given by the cumulative distribution function (CDF) for the largest eigenvalue of a complex noncentral Wishart matrix. This CDF has previously been expressed as a determinant whose elements are integrals of a confluent hypergeometric function. For the determinant elements, conventional evaluation approaches, e.g., truncation of infinite series ensuing from the hypergeometric function or numerical integration, can be unreliable and slow even for moderate antenna numbers and Rician $ K $-factor values. Therefore, herein, we derive by hand and by computer algebra also differential equations that are then solved from initial conditions computed by conventional approaches. This is the holonomic gradient method (HGM). Previous HGM-based evaluations of MIMO relied on differential equations that were not theoretically guaranteed to converge, and, thus, yielded reliable results only for few antennas or moderate $ K $. Herein, we reveal that gauge transformations can yield differential equations that are {\emph{stabile}}, i.e., guarantee HGM convergence. The ensuing HGM-based CDF evaluation is demonstrated reliable, accurate, and expeditious in computing the MRC outage probability even for very large antenna numbers and values of $ K $.

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↗
arXiv2015

$A$-Hypergeometric Distributions and Newton Polytopes

We give a bijection between a quotient space of the parameters and the space of moments for any $A$-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method to solve a conditional maximum likelihood estimation problem in statistics. Our interplay between the theory of hypergeometric functions and statistics gives some new formulas of $A$-hypergeometric polynomials.

math.CA↗
arXiv2014

Pfaffian Systems of A-Hypergeometric Equations I: Bases of Twisted Cohomology Groups

This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. Gröbner deformations give bases. These bases also give those for twisted cohomology groups. For hypergeometric system associated to a class of order polytopes, these bases have a combinatorial description. The size of the bases associated to a subclass of the order polytopes have the growth rate of the polynomial order. Bases associated to two chain posets and bouquets are studied.

math.CA↗
arXiv2013

Irregular Modified $A$-Hypergeometric Systems

A modified $A$-hypergeometric system is a system of differential equations for the function $f(t^w \cdot x)$ where $f(y)$ is a solution of an $A$-hypergeometric system in $n$ variables and $w$ is an $n$ dimensional integer vector, which is called the weight vector. We study the irregularity of modified systems by adapting to this case the notion of umbrella introduced by M. Schulze and U. Walther. Especially, we study slopes and Gevrey series solutions. We develop some applications of this study. Under some conditions we give Laplace integral representations of divergent series solutions of the modified system and we show that certain Gevrey series solutions of the original $A$-hypergeometric system along coordinate varieties are Gevrey asymptotic expansions of holomorphic solutions of the $A$-hypergeometric system.

math.CA↗
arXiv2013

Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $d$-dimensional Sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher-Bingham distribution on a $d$-dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension $d=7$ with a specified accuracy.

math.ST↗