Lecture 9: Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport

2021 ◽  
pp. 87-94
Author(s):  
Luigi Ambrosio ◽  
Elia Brué ◽  
Daniele Semola
Keyword(s):  
Metric Spaces ◽  
Optimal Transport ◽  
Dynamic Formulation

Optimal Transport in Systems and Control

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Yongxin Chen ◽  
Tryphon T. Georgiou ◽  
Michele Pavon
Keyword(s):  
Stochastic Control ◽  
Metric Spaces ◽  
Optimal Transport ◽  
Probability Distributions ◽  
Autonomous Systems ◽  
Annual Review ◽  
Publication Date ◽  
Inference Problem ◽  
Erwin Schrödinger ◽  
Variational Framework

Optimal transport began as the problem of how to efficiently redistribute goods between production and consumers and evolved into a far-reaching geometric variational framework for studying flows of distributions on metric spaces. This theory enables a class of stochastic control problems to regulate dynamical systems so as to limit uncertainty to within specified limits. Representative control examples include the landing of a spacecraft aimed probabilistically toward a target and the suppression of undesirable effects of thermal noise on resonators; in both of these examples, the goal is to regulate the flow of the distribution of the random state. A most unlikely link turned up between transport of probability distributions and a maximum entropy inference problem posed by Erwin Schrödinger, where the latter is seen as an entropy-regularized version of the former. These intertwined topics of optimal transport, stochastic control, and inference are the subject of this review, which aims to highlight connections, insights, and computational tools while touching on quadratic regulator theory and probabilistic flows in discrete spaces and networks. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

scholarly journals Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces

2019 ◽  
Vol 25 ◽  
pp. 62 ◽  
Author(s):  
Augusto Gerolin ◽  
Anna Kausamo ◽  
Tapio Rajala
Keyword(s):  
Density Functional Theory ◽  
Strong Interaction ◽  
Duality Theory ◽  
Density Functional ◽  
Metric Spaces ◽  
Optimal Transport ◽  
Transport Problem ◽  
Cost Functions ◽  
Functional Theory

In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context of SCE Density Functional Theory introduced in Strong-interaction limit of density-functional theory by Seidl [Phys. Rev. A 60 (1999) 4387].

scholarly journals Linear extension operators between spaces of Lipschitz maps and optimal transport

2020 ◽  
Vol 2020 (764) ◽  
pp. 1-21 ◽  
Author(s):  
Luigi Ambrosio ◽  
Daniele Puglisi
Keyword(s):  
Linear Extension ◽  
Metric Spaces ◽  
Optimal Transport ◽  
Partition Of Unity ◽  
Random Projection ◽  
Random Projections ◽  
Transport Distance ◽  
Lipschitz Maps ◽  
Necessary And Sufficient ◽  
Continuous Operators

AbstractMotivated by the notion of {K\hskip-0.284528pt}-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95] and the notion of {K\kern-0.284528pt}-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425], we study a weaker notion related to the Kantorovich–Rubinstein transport distance that we call {K\kern-0.284528pt}-random projection. We show that {K\kern-0.284528pt}-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak{{}^{*}} continuous operators. Finally, we use this notion to characterize the metric spaces {(X,d)} such that the free space {\mathcal{F}(X)} has the bounded approximation propriety.

scholarly journals Semi-Supervised Optimal Transport for Heterogeneous Domain Adaptation

2018 ◽  
Author(s):  
Yuguang Yan ◽  
Wen Li ◽  
Hanrui Wu ◽  
Huaqing Min ◽  
Mingkui Tan ◽  
...  
Keyword(s):  
Metric Spaces ◽  
Optimal Transport ◽  
Domain Adaptation ◽  
Learning Performance ◽  
Target Domain ◽  
Source Domain ◽  
Feature Spaces ◽  
Label Information ◽  
Real World Datasets ◽  
Heterogeneous Source

Heterogeneous domain adaptation (HDA) aims to exploit knowledge from a heterogeneous source domain to improve the learning performance in a target domain. Since the feature spaces of the source and target domains are different, the transferring of knowledge is extremely difficult. In this paper, we propose a novel semi-supervised algorithm for HDA by exploiting the theory of optimal transport (OT), a powerful tool originally designed for aligning two different distributions. To match the samples between heterogeneous domains, we propose to preserve the semantic consistency between heterogeneous domains by incorporating label information into the entropic Gromov-Wasserstein discrepancy, which is a metric in OT for different metric spaces, resulting in a new semi-supervised scheme. Via the new scheme, the target and transported source samples with the same label are enforced to follow similar distributions. Lastly, based on the Kullback-Leibler metric, we develop an efficient algorithm to optimize the resultant problem. Comprehensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of our proposed method.

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

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