Tomonari Sei

Holonomic gradient descent and its application to the Fisher-Bingham integral

We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere

Properties and applications of Fisher distribution on the rotation group

S^n

Calculating the normalising constant of the Bingham distribution on the sphere using the holonomic gradient method

, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.

Hierarchical subspace models for contingency tables

We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011) [16], and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.

A Jacobian Inequality for Gradient Maps on the Sphere and Its Application to Directional Statistics

In this paper we implement the holonomic gradient method to exactly compute the normalising constant of Bingham distributions. This idea is originally applied for general Fisher–Bingham distributions in Nakayama et al. (Adv. Appl. Math. 47:639–658, 2011). In this paper we explicitly apply this algorithm to show the exact calculation of the normalising constant; derive explicitly the Pfaffian system for this parametric case; implement the general approach for the maximum likelihood solution search and finally adjust the method for degenerate cases, namely when the parameter values have multiplicities.

Perturbation method for determining the group of invariance of hierarchical models

For the statistical analysis of multiway contingency tables, we propose modeling interaction terms in each maximal compact component of a hierarchical model. By this approach we can search for parsimonious models with smaller degrees of freedom than the usual hierarchical model, while preserving the localization property of the inference in the hierarchical model. This approach also enables us to evaluate the localization property of a given log-affine model. We discuss estimation and exact tests of the proposed model and illustrate the advantage of the proposed modeling with some data sets.

Cones of Elementary Imsets and Supermodular Functions: A Review and Some New Results

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this article, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is the sphere and the cost function is the distance squared. As an application to statistics, a new family of probability densities on the sphere is defined in terms of cost-convex functions. The log-concave property of the likelihood function follows from the inequality.

A structural model on a hypercube represented by optimal transport

We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in [S. Aoki, A. Takemura, The largest group of invariance for Markov bases and toric ideals, J. Symbolic Comput. 43 (5) (2008) 342-358]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof by Aoki and Takemura which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.

Software Packages for Holonomic Gradient Method

In this paper we give a review of the method of imsets introduced by Studeny (2005) from a geometric point of view. Elementary imsets span a polyhedral cone and its dual cone is the cone of supermodular functions. We review basic facts on the structure of these cones. Then we derive some new results on the following topics: i) extreme rays of the cone of standardized supermodular functions, ii) faces of the cones, iii) small relations among elementary imsets, and iv) some computational results on Markov basis for the toric ideal defined by elementary imsets.

An objective general index for multivariate ordered data

We propose a flexible statistical model for high-dimensional quantitative data on a hypercube. Our model, called the structural gradient model (SGM), is based on a one-to-one map on the hypercube that is a solution for an optimal transport problem. As we show with many examples, SGM can describe various dependence structures including correlation and heteroscedasticity. The maximum likelihood estimation of SGM is effectively solved by the determinant-maximization programming. In particular, a lasso-type estimation is available by adding constraints. SGM is compared with graphical Gaussian models and mixture models.