Related papers: Geometric distance between positive definite matri…

With the help of a given distance matrix of size $n$, we construct an infinite family of distances $d_p$ (where $p \geq 2$) on the complex projective space $\mathbb{P}(\mathbb{C}^n)$ modelling the space of pure states of an $n$-level…

We introduce the manifold of {\it restricted} $n\times n$ positive semidefinite matrices of fixed rank $p$, denoted $S(n,p)^{*}$. The manifold itself is an open and dense submanifold of $S(n,p)$, the manifold of $n\times n$ positive…

We study metric properties of symmetric divergences on Hermitian positive definite matrices. In particular, we prove that the square root of these divergences is a distance metric. As a corollary we obtain a proof of the metric property for…

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the…

The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We…

We review the relations between distance matrices and isometric embeddings and give simple proofs that distance matrices defined on euclidean and spherical spaces have all eigenvalues except one non-negative. Several generalizations are…

We study quantum statistical inference tasks of hypothesis testing and their canonical variations, in order to review relations between their corresponding figures of merit---measures of statistical distance---and demonstrate the crucial…

Log-Euclidean distances are commonly used to quantify the similarity between positive definite matrices using geometric considerations. This paper analyzes the behavior of this distance when it is used to measure closeness between…

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by H\"uper et al. (2021), which…

Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given…

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping $\phi:\Gamma\to\Gamma'$ between graphs from this class is shown to be an isomorphism provided that the following…

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and…

We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely ${\cal DD}_n^*$ (resp., ${\cal SDD}_n^*$), as an approximation of the semidefinite cone. We prove that the norm…

A Euclidean distance matrix $D(\alpha)$ is defined by $D_{ij}=(\alpha_i-\alpha_j)^2$, where $\alpha=(\alpha_1,\ldots,\alpha_n)$ is a real vector. We prove that $D(\alpha)$ cannot be written as a sum of $\left[2\sqrt{n}-2\right]$ nonnegative…

We show that there is an isometry between the real ambient space of all Mueller matrices and the space of all Hermitian matrices which maps the Mueller matrices onto the positive semidefinite matrices. We use this to establish an optimality…

Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$.…

This work considers the problem of estimating the distance between two covariance matrices directly from the data. Particularly, we are interested in the family of distances that can be expressed as sums of traces of functions that are…

Matrix geometric means between two positive definite matrices can be defined from distinct perspectives - as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain…

Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and…

We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is…