Related papers: The Wasserstein space of stochastic processes
Researchers from different areas have independently defined extensions of the usual weak convergence of laws of stochastic processes with the goal of adequately accounting for the flow of information. Natural approaches are convergence of…
Causal optimal transport and adapted Wasserstein distance have applications in different fields from optimization to mathematical finance and machine learning. The goal of this article is to provide equivalent formulations of these concepts…
A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and…
The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several…
The adapted Wasserstein distance is a metric for quantifying distributional uncertainty and assessing the sensitivity of stochastic optimization problems on time series data. A computationally efficient alternative to it, is provided by the…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
We investigate barycenters of Gaussian process laws in adapted Wasserstein space. The adapted Wasserstein distance refines classical optimal transport by enforcing compatibility of transport plans with the temporal flow of information, and…
We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on…
In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular,…
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…
Nowadays stochastic computer simulations with both numeral and distribution inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This paper studies the design and analysis issues of such computer…
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 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,…
Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures (i.e., the Wasserstein space). This objective is typically a divergence…
We use Wasserstein distances to characterize and study probabilistic frames. Adapting results from Olkin and Pukelsheim, from Gelbrich and from Cuesta-Albertos, Matran-Bea and Tuero-Diaz to frame operators, we show that the sets of…
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization…
The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain…
We propose a transfer principle to study the adapted 2-Wasserstein distance between stochastic processes. First, we obtain an explicit formula for the distance between real-valued mean-square continuous Gaussian processes by introducing the…
Information is an inherent component of stochastic processes and to measure the distance between different stochastic processes it is not sufficient to consider the distance between their laws. Instead, the information which accumulates…
We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the $L^2$-Wasserstein metric tensor in the probability density space to a parameter space, equipping the…