The approximation is done by minimizing the Kantorovich鈥揜ubinstein distance between distributions. Positions and probabilities of atoms of the approximating distribution are optimized. The algorithm solves a sequence of optimization problems reducing the distance between distributions. We conducted a case ...
Optimal Couplings of Kantorovich-Rubinstein-Wasserstein Lp-distance pWe achieve that the optimal solutions according to Kantorovich-Rubinstein-Wasserstein Lp-distance (p gt; 2) (abbreviation:br /KRW Lp-distance) in a bounded region of Euclidean plane satisfy a partial differential equation. We can als...
Then the Wasserstein distance is defined asW(pr,pθ)=infγ∈π∬x,y‖x−y‖γ(x,y)dxdy=infγ∈πEx,y∼γ[‖x−y‖].If we add suitable terms, we can remove all constraints on the distribution γ. This is done by adding an additional optimization over a function f:x↦k...
An easy consequence of Kantorovich-Rubinstein duality is the following: if f:[0,1]d→R is Lipschitz and {x1,…,xN}[0,1]d, then|∫[0,1]df(x)dx1N∑k=1Nf(xk)|≤‖f‖L∞W1(1N∑k=1Nδxk,dx), where W1 denotes the 1Wasserstein (or Earth Mover's) Distance. We prove a similar...
pWe achieve that the optimal solutions according to Kantorovich-Rubinstein-Wasserstein Lp-distance (p gt; 2) (abbreviation:br /KRW Lp-distance) in a bounded region of Euclidean plane satisfy a partial differential equation. We can also obtainbr /the similar results about Monge-Kantorovich problem ...
Christoph ThlearXivL. Decreusefond, M. Schulte, and C. Tha¨le, Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry, ArXiv e-prints (June 2014), available at 1406.5484....
Kantorovich–Rubinstein Distance and Barycenter for Finitely Supported Measures: Foundations and Algorithmsdoi:10.1007/s00245-022-09911-xApplied Mathematics & Optimization - The purpose of this paper is to provide a systematic discussion of a generalized barycenter based on a variant of unbalanced optimal...
distance functionWe showthat the cyclohedron (Bott-Taubes polytope)W_n arises as the polar dual of a Kantorovich-Rubinstein polytope K R(蟻), where 蟻 is an explicitly described quasi-metric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this ...
We demonstrate that this Kantorovich鈥揜ubinstein distance and extensions incorporating uncertainty in the sample locations can be written as a readily computable integral over the tree, we develop Lp Zolotarev-type generalizations of the metric, and we show how the p-value of the resulting natural...
Kantorovich-Rubinstein polytopesLipschitz polytopeCyclohedronNestohedronUnimodular triangulationsMetric spacesWe show that the cyclohedron (Bott-Taubes polytope) $W_n$ arises as the dual of a Kantorovich-Rubinstein polytope $KR(ho)$, where $ho$ is a quasi-metric (asymmetric distance function) ...