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

Abstract

We investigate a geometric computational framework, called the “scaling-rotation framework”, on $\mathrm{Sym}^{+}(p)$, 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 $\mathrm{Sym}^{+}(p)$, defined by eigenvalue multiplicities, and fibers of the “eigen-composition” map $SO(p)\times\mathrm{Diag}^{+}(p)\to\mathrm{Sym}^{+}(p)$. 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 $\mathrm{Sym}^{+}(p)$. Our main goal in this paper is the systematic characterization and analysis of minimal smooth scaling-rotation (MSSR) curves, images in $\mathrm{Sym}^{+}(p)$ of minimal-length geodesics connecting two fibers in the “upstairs” space $SO(p)\times\mathrm{Diag}^{+}(p)$. 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 the set ${\mathcal{M}}(X,Y)$ of MSSR curves from $X$ to $Y$ 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 $p=2$.