Radial fields on the manifolds of symmetric positive definite matrices

TL;DR

This paper derives explicit formulas for radial fields on the manifolds of symmetric positive definite matrices, which are crucial for statistical analysis and geometric understanding of these spaces.

Contribution

It provides the first explicit expression for radial fields on SPD manifolds and proves their smoothness, filling a key gap in geometric analysis.

Findings

Derived explicit formulas for radial fields on SPD manifolds.

Proved the smoothness of these radial fields.

Facilitated potential applications in statistics and data analysis.

Abstract

On Hadamard manifolds, the radial fields, which are the negative gradients of the Busemann functions, can be used to designate a canonical sense of direction. This could have many potential applications to Hadamard manifold-valued data, for example in defining notions of quantiles or treatment effects. Some of the most commonly encountered Hadamard manifolds in statistics are the spaces of symmetric positive definite matrices, which are used in, for example, covariance matrix analysis and diffusion tensor imaging. Surprisingly, an expression for the radial fields on these manifolds is unavailable in the literature even though the issue arises quite naturally when studying the geometry of these spaces. This paper aims to fill this gap by deriving such an expression, and also demonstrates their smoothness.