References for the Holonomic Gradient Method (HGM) and
the Holonomic Gradient Descent Method (HGD)
Papers and Tutorials
- T.Koyama,
Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,
arxiv:1311.6905
- T.Hibi, K.Nishiyama, N.Takayama,
Pfaffian Systems of A-Hypergeometric Equations I,
Bases of Twisted Cohomology Groups,
arxiv:1212.6103
(major revision v2 of arxiv:1212.6103)
-
T.Hibi et al, Groebner Bases : Statistics and Software Systems , Springer, 2013.
-
Introduction to the Holonomic Gradient Method (movie), 2013.
movie at youtube
- T.Sei, A.Kume,
Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method,
Statistics and Computing, 2013,
DOI
- T.Koyama, A.Takemura,
Calculation of Orthant Probabilities by the Holonomic Gradient Method,
arxiv:1211.6822
- T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Holonomic Rank of the Fisher-Bingham System of Differential Equations,
to appear in Journal of Pure and Applied Algebra
-
T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Holonomic Gradient Descent for the Fisher-Bingham Distribution on the d-dimensional Sphere,
Computational Statistics (2013)
DOI
- Hiroki Hashiguchi, Yasuhide Numata, Nobuki Takayama, Akimichi Takemura,
Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
Journal of Multivariate Analysis, 117, (2013) 296-312,
DOI
- Tomonari Sei, Hiroki Shibata, Akimichi Takemura, Katsuyoshi Ohara, Nobuki Takayama,
Properties and applications of Fisher distribution on the rotation group,
Journal of Multivariate Analysis, 116 (2013), 440--455,
DOI
- T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral,
Funkcialaj Ekvacioj 56 (2013), 51--61.
jstage
-
Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,
Tomonari Sei, Nobuki Takayama, Akimichi Takemura ,
Holonomic Gradient Descent and its Application to Fisher-Bingham Integral,
Advances in Applied Mathematics 47 (2011), 639--658,
DOI
Three Steps of HGM
- Find a holonomic system satisfied by the normalizing constant.
We may use computational or theoretical methods to find it.
Groebner basis and related methods are used.
- Find an initial value vector for the holonomic system.
This is equivalent to evaluating the normalizing constant and its derivatives
at a point.
This step is usually performed by a series expansion.
- Solve the holonomic system numerically. We use several methods
in numerical analysis such as the Runge-Kutta method of solving
ordinary differential equations and efficient solvers of systems of linear
equations.
Software Packages for HGM
- hgm package for R for the step 3.
- yang (for Pfaffian systems) , nk_restriction (for D-module integrations),
tk_jack (for Jack polynomials) are for the steps 1 or 2 and in the
asir-contrib
Programs to try examples of our papers
- d-dimensional Fisher-Bingham System
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