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 Sym + (p), the set of p × 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 Sym + (p), defined by eigenvalue multiplicities, and fibers of the "eigencomposition" map SO(p) × Diag + (p) → 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 Sym + (p). Our main goal in this paper is the systematic characterization and analysis of minimal smooth scalingrotation (MSSR) curves, images in Sym + (p) of minimal-length geodesics connecting two fibers in the "upstairs" space SO(p)×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 lowdimensional case p = 3 (the home of DTI), we find new explicit formulas for MSSR curves and for the scaling-rotation distance, and identify 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 scalingrotation distance and MSSR curves for the case p = 2.MSC 2010 subject classifications: Primary 53C99; secondary 53C15, 53C22, 51F25, 15A18. Groisser et al./Eigenvalue stratification and MSSR curves(For any subgroup H ⊂ O(p), we write S(H) for H SO(p).) In this example, G J has two connected components, one in which det(R 1 ) = det(R 2 ) = 1 (the component G 0 J ), and one in which det(R 1 ) = det(R 2 ) = −1. For general J = {J 1 , . . . , J r }, the elements of G J have "interleaved blocks". Writing k i = |J i |, we have(2.4) and the identity component G 0 J is isomorphic to SO(k 1 ) × SO(k 2 ) · · · × SO(k r ). If the k i are non-decreasing then [J] = k 1 + · · · + k r . For concreteness we define(2.5)The groups G J are also partially-ordered. For J, K ∈ Part({1, . . . , p}),This partial-ordering will be reflected in the stratifications of Sym + (p) and M discussed in Section 2.7.Definition 2.4 For each D ∈ Diag(p), we define the stabilizer group of D,

Stochastic modelling of symmetric positive-definite material tensors

“…the eigenvalues of the generalised eigenvalue problem C 1 s k = λ k C 2 s k . This distance satisfies all the desired properties [1,35,36] listed in Section 2, as well as that the boundary ∂ Sym + (d) is infinitely far away from the identity-this is here a consequence of working with log C. But other choices for the metric are possible [15,16]. Observe that when C 1 and C…”

“…Various options are explored e.g. in [36], see also the discussion in [16]. The open convex cone Sym + (d) ⊂ Sym(d) may be considered as a metric space in ways different from just being a subset of the vector space Sym(d).…”

“…From this one may come to the idea [24,16] to measure the distance on Sym + (d) by measuring separately the distance between the eigenspaces (rotation) and the distance between the eigenvalues (scaling), and one may define a distance function (not always a metric [16]) 2 for some c > 0, i.e. measure the distance between the scales and the orientations separately and add them both up.…”

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

“…The uniqueness-related results in this paper contribute to a rigorous and systematic description of the geometry and topology of the triple (M (p), F, Sym + (p)), and to a firm foundation for further study of the scalingrotation framework, such as in [7] and [8].…”

“…Appendix A provides a thorough picture of the fiber structure of M (p), including its inextricable tie to the group S+ p of "even signed-permutations", a group not to be confused with a more familiar group of the same order and similar-sounding description in terms of signs and permutations, the Weyl group of the simple Lie algebra D p . Some results proven in Appendix A are applied earlier in the main body of this paper, and some were previously stated without proof in [7] and applied there.…”