Related papers: Shape-constrained density estimation with Wasserst…
This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…
Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between L\'{e}vy measures to solve a…
We investigate the stability of the Wasserstein distance, a metric structure on the space of probability measures arising from the theory of optimal transport, under metric ultralimits. We first show that if $(X_{i},d_{i})_{i\in\mathbb{N}}$…
We introduce and analyse a new nonparametric estimator of a multi-dimensional density. Our smooth projection estimator (SPE) is defined by a least squares projection of the sample onto an infinite dimensional mixture class via an…
Computing the infinity Wasserstein distance and retrieving projections of a probability measure onto a closed subset of probability measures are critical sub-problems in various applied fields. However, the practical applicability of these…
Data consisting of time-indexed distributions of cross-sectional or intraday returns have been extensively studied in finance, and provide one example in which the data atoms consist of serially dependent probability distributions.…
The question of optimally approximating an arbitrary probability measure in the Wasserstein distance by a discrete one with uniform weights is considered. Estimates are obtained for the optimal approximation distance, with an explicit rate…
In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $p\geq1$, and for densities with unknown regularity measured on a Besov scale.…
Optimal transport has found widespread applications in signal processing and machine learning. Among its many equivalent formulations, optimal transport seeks to reconstruct a random variable/vector with a prescribed distribution at the…
The Sliced-Wasserstein distance (SW) is being increasingly used in machine learning applications as an alternative to the Wasserstein distance and offers significant computational and statistical benefits. Since it is defined as an…
Chance constraints yield non-convex feasible regions in general. In particular, when the uncertain parameters are modeled by a Wasserstein ball, arXiv:1806.07418 and arXiv:1809.00210 showed that the distributionally robust (pessimistic)…
We study several constrained variational problem in the 2-Wasserstein metric for which the set of probability densities satisfying the constraint is not closed. For example, given a probability density $F_0$ on $\R^d$ and a time-step $h>0$,…
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear…
Shape-constrained inference has wide applicability in bioassay, medicine, economics, risk assessment, and many other fields. Although there has been a large amount of work on monotone-constrained univariate curve estimation, multivariate…
Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally…
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…
We study the estimation problem of distribution-on-distribution regression, where both predictors and responses are probability measures. Existing approaches typically rely on a global optimal transport map or tangent-space linearization,…
Optimization over the space of probability measures endowed with the Wasserstein-2 geometry is central to modern machine learning and mean-field modeling. However, traditional methods relying on full Wasserstein gradients often suffer from…
Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the…
In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the…