Warped geometries of Segre-Veronese manifolds

TL;DR

This paper introduces a family of warped geometries on Segre-Veronese manifolds, providing explicit formulas for key geometric maps, and explores their properties and applications in tensor averaging.

Contribution

It presents a novel one-parameter family of warped geometries on Segre-Veronese manifolds with explicit formulas for exponential and logarithmic maps.

Findings

Segre-Veronese manifolds are not geodesically connected in Euclidean geometry.

Warped geometries can ensure geodesic connectedness for certain parameters.

Application demonstrated in computing the Riemannian center of mass for tensors.

Abstract

Segre-Veronese manifolds are smooth submanifolds of tensors comprising the partially symmetric rank-1 tensors. We investigate a one-parameter family of warped geometries of Segre-Veronese manifolds, which includes the standard Euclidean geometry. This parameter controls by how much spherical tangent directions are weighted relative to radial tangent directions. We present closed expressions for the exponential map, the logarithmic map, and the intrinsic distance on these warped Segre-Veronese manifolds, which can be computed efficiently numerically. It is shown that Segre-Veronese manifolds are not geodesically connected in the Euclidean geometry, while they are for some values of the warping parameter. The benefits of geodesics connectedness may outweigh using the Euclidean geometry in certain applications. One such application is presented: numerically computing the Riemannian center of mass for averaging rank-1 tensors.