English

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

R2 v1 2026-06-21T13:09:34.427Z