Related papers: The Linearized Hellinger--Kantorovich Distance
Given a complex structure $J$ on a real (finite or infinite dimensional) Hilbert space $H$, we study the geometry of the Lagrangian Grassmannian $\Lambda(H)$ of $H$, i.e. the set of closed linear subspaces $L\subset H$ such that…
In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted…
The goal of this note is to demonstrate how existing results can be adapted to establish the following result: A locally metric measure homogeneous $\mathrm{RCD}(K,N)$ space is isometric to, after multiplying a positive constant to the…
On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of…
This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are…
Suppose $X$ is an $\rm{RCD}(K,N)$ space with $K \in \mathbb{R}$ and $N \in (1,\infty)$. We obtain a characterisation of the Newtonian-Sobolev space $N^{1,2}(X)$ in terms of a quantity which measures to what extent a function is locally…
The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the standard Euclidean distance or…
This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are…
In the framework of quasi-Hermitian quantum mechanics it is shown that a weakening of the isotropy of the Hilbert-space geometry can help us to enlarge the domain of the parameters at which the evolution is unitary. The idea is tested using…
In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings.…
A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved,…
Trans-Planckian redshifts in cosmology and outside black holes may provide windows on a hypothetical short distance cutoff on the fundamental degrees of freedom. In cosmology, such a cutoff seems to require a growing Hilbert space, but for…
This work introduces a new, explicit bound on the Hellinger distance between a continuous random variable and a Gaussian with matching mean and variance. As example applications, we derive a quantitative Hellinger central limit theorem and…
In this paper, we develop the foundations of the theory of quasiregular mappings in general metric measure spaces. In particular, nine definitions of quasiregularity for a discrete open mapping with locally bounded multiplicity are proved…
Parametric models in vector spaces are shown to possess an associated linear map. This linear operator leads directly to reproducing kernel Hilbert spaces and affine- / linear- representations in terms of tensor products. From the…
The construction of $r$-nets offers a powerful tool in computational and metric geometry. We focus on high-dimensional spaces and present a new randomized algorithm which efficiently computes approximate $r$-nets with respect to Euclidean…
We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich…
We derive a canonical form for smooth vector fields on $\Re^{n+1}$. We use this to demonstrate the local multi-Hamiltonian nature of the corresponding flows. Associated with the canonical form is an inhomogenious linear PDE whose solutions…
Motivated by the approach of random linear codes, a new distance in the vector space over a finite field is defined as the logarithm of the "surface area" of a Hamming ball with radius being the corresponding Hamming distance. It is named…
The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. First, we show the existence of a good product-state approximation for the ground-state energy of…