Geometric statistics with subspace structure preservation for SPD matrices

TL;DR

The paper addresses SPD-valued data processing while preserving subspace structure, challenging with traditional affine-invariant Riemannian geometry. It proposes a Thompson-geometry framework on the semidefinite cone, using extreme generalized eigenvalues to define efficient interpolations and an inductive mean. Key results include a span-preserving Thompson geodesic $X*_tY$, determinant evolution demonstrating shrinkage along Thompson interpolations, explicit bounds on mid-point distances, and a sparsity- and affine-equivariant inductive mean. These contributions enable scalable, structure-aware processing of large, sparse SPD matrices in applications such as diffusion tensor imaging and kernel methods.

Abstract

We present a geometric framework for the processing of SPD-valued data that preserves subspace structures and is based on the efficient computation of extreme generalized eigenvalues. This is achieved through the use of the Thompson geometry of the semidefinite cone. We explore a particular geodesic space structure in detail and establish several properties associated with it. Finally, we review a novel inductive mean of SPD matrices based on this geometry.

Figures (6)

• Figure 1: Reconstructions of the middle layer of an image consisting of three layers of ellipsoids representing diffusion tensors using Euclidean, Riemannian, and Thompson geodesic interpolations. The Euclidean interpolation exhibits an undesirable swelling effect. The Riemannian and Thompson interpolations generate similar results and are more faithful to the original data.
• Figure 2: Plots of the logarithm of the determinants along Euclidean, Riemannian, and Thompson geodesics connecting 1,000 pairs of randomly generated $n\times n$ SPD matrices of unit determinant for $n=3$ and $n=10$. We observe that the determinants along the Riemannian geodesics are constant and equal to one as expected. The determinants along Euclidean interpolations exhibit swelling, whereas an opposite shrinkage effect is observed for interpolations along the Thompson geodesics. Both of these effects are magnified in higher dimensions.
• Figure 3: Plots of $f(\boldsymbol{\lambda})$ (\ref{['f']}) against $r=\max_i|\log\lambda_i|$ for 100,000 samples $\boldsymbol{\lambda}\in\mathbb{R}^n_+$ for $n=4$ and $n=20$. The solid blue curves represent the upper bounds of $f(\boldsymbol{\lambda})$ given in (\ref{['f bounds']}). The color scheme is used to indicate the density of plot points $(r,f(\boldsymbol{\lambda}))$.
• Figure 4: Generate inductive sequence of SPD matrices $(X_i)_{i\geq 1}$ from an initial point $X_1$ and the finite ordered set $\mathcal{P}=(Y_1,\cdots,Y_k)\subset\mathbb{S}^n_{++}$
• Figure 5: The inductive Thompson mean and Riemannian mean for a set of 5 input data SPD matrices with the same sparsity pattern.