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
Most software packages are experimental and temporary documents are found in
"asir-contrib manual" (auto-autogenerated part), or
"Experimental Functions in Asir", or "miscellaneous and other documents"
of the
OpenXM documents.
The nightly snapshot of the asir-contrib can be found in the Asir-Contrib page below,
or look up our
cvsweb page
- hgm package for R for the step 3.
- yang (for Pfaffian systems) , nk_restriction (for D-module integrations),
tk_jack (for Jack polynomials), ko_fb_pfaffian (Pfaffian system for the Fisher-Bingham system)
are for the steps 1 or 2 and in the
asir-contrib
- nk_fb_gen_c is a package to generate a C program to perform
maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent)
It is in the
asir-contrib
Programs to try examples of our papers
- d-dimensional Fisher-Bingham System
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