# Is affine invariance well defined on SPD matrices? A principled continuum of metrics

## TL;DR

This paper explores the theoretical landscape of metrics on SPD matrices, introducing a continuum of affine-invariant metrics and examining principles to guide metric selection for applications.

## Contribution

It introduces a continuum of affine-invariant metrics on SPD matrices, including power-affine and deformed-affine metrics, and investigates principles for selecting appropriate metrics.

## Key findings

- Introduces a continuum of metrics on SPD matrices.
- Analyzes principles guiding metric choice.
- Provides theoretical insights into affine invariance.

## Abstract

Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the choice of one particular metric for a given application. Invariance under the action of the affine transformations was suggested as a principle. Another concept is based on symmetries. However, the affine-invariant metric and the recently proposed polar-affine metric are both invariant and symmetric. Comparing these two cousin metrics leads us to introduce much wider families: power-affine and deformed-affine metrics. Within this continuum, we investigate other principles to restrict the family size.