相关论文: Computing Kantorovich-Wasserstein Distances on $d$…
This paper continues the investigation of `Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N…
We propose a new algorithm that uses an auxiliary neural network to express the potential of the optimal transport map between two data distributions. In the sequel, we use the aforementioned map to train generative networks. Unlike WGANs,…
The notions of cutwidth and pathwidth of digraphs play a central role in the containment theory for tournaments, or more generally semi-complete digraphs, developed in a recent series of papers by Chudnovsky, Fradkin, Kim, Scott, and…
Optimal Transport (OT) metrics allow for defining discrepancies between two probability measures. Wasserstein distance is for longer the celebrated OT-distance frequently-used in the literature, which seeks probability distributions to be…
We propose a discrete transport equation on graphs which connects distributions on both vertices and edges. We then derive a discrete analogue of the Benamou-Brenier formulation for Wasserstein-$1$ distance on a graph and as a result…
We consider the optimal transportation problem on a globally hyperbolic spacetime for some cost function $c_2$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is the…
In this thesis I propose an algorithm to heuristically calculate different distance measures on uncertain graphs (i.e. graphs where edges only exist with a certain probability) and apply this to the heuristic calculation of harmonic…
We discuss the classical problem of measuring the regularity of distribution of sets of $N$ points in $\mathbb{T}^d$. A recent line of investigation is to study the cost ($=$ mass $\times$ distance) necessary to move Dirac measures placed…
A representation for the Kantorovich--Rubinstein distance between probability measures on an abstract Wiener space in terms of the extended stochastic integral (or, divergence) operator is obtained.
In this paper, we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph $G$. Different graph structures lead to different specifications of the optimal transport problem. For…
Wasserstein distances are widely used in modern data analysis but pose significant computational and statistical challenges in high dimensions. The sliced Wasserstein distance alleviates these challenges by leveraging one-dimensional…
We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is…
We study dynamical optimal transport of discrete time systems (dDOT) with Lagrangian cost. The problem is approached by combining optimal control and Kantorovich duality theory. Based on the derived solution, a first order splitting…
The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject…
We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic…
We study a multi-marginal optimal transportation problem with a cost function of the form $c(x_{1}, \ldots,x_{m})=\sum_{k=1}^{m-1}|x_{k}-x_{k+1}|^{2} + |x_{m}- F(x_{1})|^{2}$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$. When $m=4$,…
Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal…
It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the…
This work investigates several aspects related to quantitative stability in optimal transport, as well as uniqueness of the dual transport problem. Our main contributions are as follows. Chapter 1: Observations regarding the quantitative…
The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…