Geometric foundations for scaling-rotation statistics on symmetric positive definite matrices: minimal smooth scaling-rotation curves in low dimensions

TL;DR

This paper develops a geometric framework for analyzing symmetric positive definite matrices, especially in low dimensions, providing explicit formulas for minimal smooth scaling-rotation curves and distances, which are useful in applications like diffusion-tensor imaging.

Contribution

The paper introduces a systematic analysis of minimal smooth scaling-rotation curves on SPD matrices, with explicit formulas for low-dimensional cases p=2 and p=3, advancing geometric statistical analysis methods.

Findings

Explicit formulas for MSSR curves in p=2 and p=3 cases.

Closed-form expressions for scaling-rotation distance.

Identification of minimal curves in all nontrivial cases for p=3.

Abstract

We investigate a geometric computational framework, called the "scaling-rotation framework", on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite (SPD) matrices. The purpose of our study is to lay geometric foundations for statistical analysis of SPD matrices, in situations in which eigenstructure is of fundamental importance, for example diffusion-tensor imaging (DTI). Eigen-decomposition, upon which the scaling-rotation framework is based, determines both a stratification of ${\rm Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map $SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. This leads to the notion of scaling-rotation distance [Jung et al. (2015)], a measure of the minimal amount of scaling and rotation needed to transform an SPD matrix, $X,$ into another, $Y,$ by a smooth curve in ${\rm Sym}^+(p)$. Our main goal in this paper is the systematic characterization and analysis of minimal smooth scaling-rotation (MSSR) curves, images in ${\rm Sym}^+(p)$ of minimal-length geodesics connecting two fibers in the "upstairs" space $SO(p)\times{\rm Diag}^+(p)$. The length of such a geodesic connecting the fibers over $X$ and $Y$ is what we define to be the scaling-rotation distance from $X$ to $Y.$ For the important low-dimensional case $p = 3$ (the home of DTI), we find new explicit formulas for MSSR curves and for the scaling-rotation distance, and identify ${\cal M}(X,Y)$ in all "nontrivial" cases. The quaternionic representation of $SO(3)$ is used in these computations. We also provide closed-form expressions for scaling-rotation distance and MSSR curves for the case $p = 2$.