Uniqueness questions in a scaling-rotation geometry on the space of symmetric positive-definite matrices

Abstract

Jung et al. (2015) introduced a geometric structure on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of ${\rm Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map $F:M(p):=SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. When $M(p)$ is equipped with a suitable Riemannian metric, the fiber structure leads to notions of scaling-rotation distance between $X,Y\in {\rm Sym}^+(p)$, the distance in $M(p)$ between fibers $F^{-1}(X)$ and $F^{-1}(Y)$, and minimal smooth scaling-rotation (MSSR) curves, images in ${\rm Sym}^+(p)$ of minimal-length geodesics connecting two fibers. In this paper we study the geometry of the triple $(M(p),F,{\rm Sym}^+(p))$, focusing on some basic questions: For which $X,Y$ is there a unique MSSR curve from $X$ to $Y$? More generally, what is the set ${\cal M}(X,Y)$ of MSSR curves from $X$ to $Y$? This set is influenced by two potential types of non-uniqueness. We translate the question of whether the second type can occur into a question about the geometry of Grassmannians $G_m({\bf R}^p)$, with $m$ even, that we answer for $p\leq 4$ and $p\geq 11$. Our method of proof also yields an interesting half-angle formula concerning principal angles between subspaces of ${\bf R}^p$ whose dimensions may or may not be equal. The general-$p$ results concerning MSSR curves and scaling-rotation distance that we establish here underpin the explicit $p=3$ results in Groisser et al. (2017). Addressing the uniqueness-related questions requires a thorough understanding of the fiber structure of $M(p)$, which we also provide.