Geometric distance between positive definite matrices of different dimensions
Provides a novel geometric framework for comparing positive definite objects of different dimensions, addressing a gap in existing Riemannian geometry methods.
The paper extends the Riemannian distance on positive definite matrices to compare matrices of different dimensions, enabling geometric distance between ellipsoids, inner products, or covariance matrices of varying sizes.
We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of $n\times n$ real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that $\mathbb{S}^n_{++}$ also parameterizes $n$-dimensional ellipsoids, and inner products on $\mathbb{R}^n$, $n \times n$ covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.