Related papers: On the transport dimension of measures

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…

In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical…

We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling---i.e., a transport map---between a complex "target" probability measure of interest and a simpler reference measure.…

This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and…

Let $\mu$ be a Borel probability measure generated by a hyperbolic recurrent iterated function system defined on a nonempty compact subset of $\mathbb R^k$. We study the Hausdorff and the packing dimensions, and the quantization dimensions…

In machine learning and computer vision, optimal transport has had significant success in learning generative models and defining metric distances between structured and stochastic data objects, that can be cast as probability measures. The…

For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We…

We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We…

This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the…

Kifer, Peres, and Weiss proved that there exists $c_0>0,$ such that $\dim \mu\leq 1-c_0$ for any probability measure $\mu$ which makes the digits of the continued fraction expansion i.i.d. random variables. In this paper we prove that…

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the…

There are interesting extensions of the problem of determining a joint probability with known marginals. On the one hand, one may impose size constraints on the joint probabilities. On the other, one may impose additional constraints like…

Transport properties of particles and waves in spatially periodic structures that are driven by external time-dependent forces manifestly depend on the space-time symmetries of the corresponding equations of motion. A systematic analysis of…

We introduce folded optimal transport, as a method to extend a cost or distance defined on the extreme boundary of a convex to the whole convex, related to convex extension. This construction broadens the framework of standard optimal…

This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density $\rho$ and a probability measure $\mu$ on R^d , which we denote T$\mu$. Assuming that the source density $\rho$ is…

Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this…

We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by $\rho \geq 1$, and provide a metric space $\mathcal{R}_{\rho}(\cdot, \cdot)$…

We introduce a novel optimal transport framework for probabilistic circuits (PCs). While it has been shown recently that divergences between distributions represented as certain classes of PCs can be computed tractably, to the best of our…

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…

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective…