Related papers: Measure dynamics with Probability Vector Fields an…
A new type of differential equations for probability measures on Euclidean spaces, called Measure Differential Equations (briefly MDEs), is introduced. MDEs correspond to Probability Vector Fields, which map measures on an Euclidean space…
We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus,…
Many scientific systems, such as cellular populations or economic cohorts, are naturally described by probability distributions that evolve over time. Predicting how such a system would have evolved under different forces or initial…
In [Gwiazda, Jamr\'oz, Marciniak-Czochra 2012] a framework for studying cell differentiation processes based on measure-valued solutions of transport equations was introduced. Under application of the so-called measure-transmission…
We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a…
Measuring dependence between random variables is a fundamental problem in Statistics, with applications across diverse fields. While classical measures such as Pearson's correlation have been widely used for over a century, they have…
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact…
Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in…
We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using…
In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of such distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized…
Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their…
This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain…
The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning. Established comparison methods treat the standard setting that the distributions are supported in the same space. Recently,…
We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight…
We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean…
We introduce a class of backward stochastic differential equations (BSDEs) on the Wasserstein space of probability measures. This formulation extends the classical correspondence between BSDEs, stochastic control, and partial differential…
Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of…
We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of…
Thermodynamics serves as a universal means for studying physical systems from an energy perspective. In recent years, with the establishment of the field of stochastic and quantum thermodynamics, the ideas of thermodynamics have been…
We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the…