A Jacobian inequality for gradient maps on the sphere and its application to directional statistics
Differential Geometry
2010-02-08 v3 Statistics Theory
Statistics Theory
Abstract
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 paper, 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. The proof uses the non-negative cross-curvature property of the sphere recently established by Kim and McCann, and Figalli and Rifford. 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.
Keywords
Cite
@article{arxiv.0906.0874,
title = {A Jacobian inequality for gradient maps on the sphere and its application to directional statistics},
author = {Tomonari Sei},
journal= {arXiv preprint arXiv:0906.0874},
year = {2010}
}
Comments
20 pages, 14 figures