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Given a probability measure with density, Fermat distances and density-driven metrics are conformal transformations of the Euclidean metric that shrink distances in high density areas and enlarge distances in low density areas. Although…
This paper focuses on a similarity measure, known as the Wasserstein distance, with which to compare images. The Wasserstein distance results from a partial differential equation (PDE) formulation of Monge's optimal transport problem. We…
We propose the Wasserstein-Fourier (WF) distance to measure the (dis)similarity between time series by quantifying the displacement of their energy across frequencies. The WF distance operates by calculating the Wasserstein distance between…
In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures…
Comparing probability distributions is at the crux of many machine learning algorithms. Maximum Mean Discrepancies (MMD) and Wasserstein distances are two classes of distances between probability distributions that have attracted abundant…
We propose two new evaluation metrics to assess realness of generated images based on normalizing flows: a simpler and efficient flow-based likelihood distance (FLD) and a more exact dual-flow based likelihood distance (D-FLD). Because…
We establish conditions to characterize probability measures by their $L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic…
In this paper we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance…
Leveraging the Wasserstein distance -- a summation of sample-wise transport distances in data space -- is advantageous in many applications for measuring support differences between two underlying density functions. However, when supports…
The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which…
A new similarity measure for two sets of S-parameters is proposed. It is constructed with the modified Hausdorff distance applied to S-parameter points in 3D space with real, imaginary and normalized frequency axes. New S-parameters…
Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected…
Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit interval. For fixed $\mathbb{P}$ the Wasserstein distance between $\hat{\mathbb{P}}_N$ and…
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
With the increasing availability of data objects in the form of probability distributions, there is a growing need for statistical methods tailored to distributional data. Distance measures, especially the pairwise distance matrix between…
We derive the sampling probability density function (pdf) of an ideal localized random electromagnetic field, its amplitude and intensity in an electromagnetic environment that is quasi-statically time-varying statistically homogeneous or…
This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is…
Generalized Wasserstein distances allow to quantitatively compare two continuous or atomic mass distributions with equal or different total mass. In this paper, we propose four numerical methods for the approximation of three different…
The Wasserstein metric or earth mover's distance (EMD) is a useful tool in statistics, machine learning and computer science with many applications to biological or medical imaging, among others. Especially in the light of increasingly…
We develop a general framework for statistical inference with the 1-Wasserstein distance. Recently, the Wasserstein distance has attracted considerable attention and has been widely applied to various machine learning tasks because of its…