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Optimal Transport Approximation of 2-Dimensional Measures [article]

Frédéric de Gournay, Jonas Kahn, Léo Lebrat, Pierre Weiss
2019 arXiv   pre-print
We propose a fast and scalable algorithm to project a given density on a set of structured measures defined over a compact 2D domain. The measures can be discrete or supported on curves for instance.  ...  The proposed principle and algorithm are a natural generalization of previous results revolving around the generation of blue-noise point distributions, such as Lloyd's algorithm or more advanced techniques  ...  between convolution and optimal transport based approximation of measures.  ... 
arXiv:1804.08356v2 fatcat:mnuz34of5ncuzacz4zmgpl2y4u

Sampling via Measure Transport: An Introduction [chapter]

Youssef Marzouk, Tarek Moselhy, Matthew Parno, Alessio Spantini
2016 Handbook of Uncertainty Quantification  
of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces.  ...  We then address practical issues associated with the optimization--based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error  ...  The structure of the optimal transport map follows not only from the target and reference measures but also from the cost function in (2) .  ... 
doi:10.1007/978-3-319-11259-6_23-1 fatcat:7nadnr463jef7hyusnzfs6726q

Exchangeable optimal transportation and log-concavity [article]

Alexander V. Kolesnikov, Danila A. Zaev
2015 arXiv   pre-print
We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution.  ...  We study the Monge and Kantorovich transportation problems on R^∞ within the class of exchangeable measures.  ...  We construct the optimal mapping as a limit of finite-dimensional approximations.  ... 
arXiv:1511.09025v1 fatcat:asaqwqp6sneefh3vntpma2lbwi

Optimal transport over nonlinear systems via infinitesimal generators on graphs

Karthik Elamvazhuthi, Piyush Grover
2018 Journal of Computational Dynamics  
The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time.  ...  Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic.  ...  Hence, we approximate solutions of optimal transport problems on an Euclidean space using solutions of optimal transport problems on graphs.  ... 
doi:10.3934/jcd.2018001 fatcat:2l6745fq3zahhjg5px6j2bqz24

Error bounds for discretized optimal transport and its reliable efficient numerical solution [article]

Sören Bartels, Stephan Hertzog
2017 arXiv   pre-print
We derive error estimates for the approximation of the problem using convex combinations of Dirac measures and devise an active-set strategy that uses the optimality conditions to predict the support of  ...  The discretization of optimal transport problems often leads to large linear programs with sparse solutions.  ...  SB acknowledges support by the DFG via the priority program Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization (SPP 1962).  ... 
arXiv:1710.04888v1 fatcat:wowcusdmpvcihn7vmlrydtcvpa

Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark [article]

Alexander Korotin, Lingxiao Li, Aude Genevay, Justin Solomon, Alexander Filippov, Evgeny Burnaev
2021 arXiv   pre-print
In this paper, we address this issue for quadratic-cost transport -- specifically, computation of the Wasserstein-2 distance, a commonly-used formulation of optimal transport in machine learning.  ...  This strategy yields pairs of continuous benchmark measures in high-dimensional spaces such as spaces of images.  ...  Background on Optimal Transport We start by stating the definition and some properties of optimal transport with quadratic cost.  ... 
arXiv:2106.01954v2 fatcat:76l2rawcxjedvcbteyrnbqd7ou

Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation

Sangeetika Ruchi, Svetlana Dubinkina, Jana de Wiljes
2021 Nonlinear Processes in Geophysics  
The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping.  ...  Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution.  ...  Figure 2 . 2 Application to F1 parameterisation: using Sinkhorn approximation (a) and optimal transport resampling (b). Box plot over 20 independent simulations of the RMSE of mean log permeability.  ... 
doi:10.5194/npg-28-23-2021 fatcat:25i5pgdjc5e7jmp45mfbuvhtty

Efficient estimates of optimal transport via low-dimensional embeddings [article]

Patric M. Fulop, Vincent Danos
2021 arXiv   pre-print
Optimal transport distances (OT) have been widely used in recent work in Machine Learning as ways to compare probability distributions.  ...  We extend this approach and show that one can approximate OT distances by using more general families of maps provided they are 1-Lipschitz.  ...  Acknowledgments The work of Patric Fulop was supported by Microsoft Research through its PhD Scolarship Programme and the University of Edinburgh.  ... 
arXiv:2111.04838v1 fatcat:p4q2qqozurhxxcnv7fdbnsrwj4

A Review on Modern Computational Optimal Transport Methods with Applications in Biomedical Research [article]

Jingyi Zhang, Wenxuan Zhong, Ping Ma
2021 arXiv   pre-print
As a powerful tool to transport between two probability measures, optimal transport methods have been reinvigorated nowadays in a remarkable proliferation of modern data science applications.  ...  Optimal transport has been one of the most exciting subjects in mathematics, starting from the 18th century.  ...  National Institute of Health under grant R01GM122080.  ... 
arXiv:2008.02995v3 fatcat:b4x4tvenhzdn3kbmawg26742bq

A data-driven linear-programming methodology for optimal transport [article]

Weikun Chen, Esteban G. Tabak
2017 arXiv   pre-print
Both the marginal distributions and their unknown optimal coupling are approximated through mixtures, which decouples the problem into the the optimal transport between the individual components of the  ...  A data-driven formulation of the optimal transport problem is presented and solved using adaptively refined meshes to decompose the problem into a sequence of finite linear programming problems.  ...  measure of a random vector X, and the optimal transportation maps between probability measure ν and ν j is denoted T j , so (22) L(T j (X)) = ν j , when L(X) = ν.  ... 
arXiv:1710.03327v1 fatcat:jnlbkfr6c5ds3ijrhgax2w26by

Brenier approach for optimal transportation between a quasi-discrete measure and a discrete measure [article]

Ying Lu, Liming Chen, Alexandre Saidi, Xianfeng Gu
2018 arXiv   pre-print
Recently, Cuturi proposed the Sinkhorn distance which makes use of an approximate Optimal Transport cost between two distributions as a distance to describe distribution discrepancy.  ...  The first one is that the Sinkhorn distance only gives an approximation of the real Wasserstein distance, the second one is the 'divide by zero' problem which often occurs during matrix scaling when setting  ...  the approximate Brenier method to learn the current optimal transportation map.  ... 
arXiv:1801.05574v1 fatcat:5qb4pnxnfvdbbbcz5qzm7hp7gu

Multiscale Strategies for Computing Optimal Transport [article]

Samuel Gerber, Mauro Maggioni
2017 arXiv   pre-print
This paper presents a multiscale approach to efficiently compute approximate optimal transport plans between point sets.  ...  An analysis of sets of brain MRI based on optimal transport distances illustrates the effectiveness of the proposed method on a real world data set.  ...  For embedding the optimal transport distance we consider two approaches: 1. A five dimensional MDS embedding based on the transport cost at the finest scale. 2.  ... 
arXiv:1708.02469v1 fatcat:6m4xzllpdfdmflxgnoh4gf7jzu

Transform-based particle filtering for elliptic Bayesian inverse problems

Sangeetika Ruchi, Svetlana Dubinkina, Marco A Iglesias
2019 Inverse Problems  
We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements.  ...  We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial  ...  Acknowledgments This work is part of the research programme Shell-NWO/FOM Computational Sciences for Energy Research (CSER) with project number 14CSER007 which is partly financed by the Netherlands Organization  ... 
doi:10.1088/1361-6420/ab30f3 fatcat:uqmpn36thrbw3n5tvtjhryruzq

Graph Recovery From Incomplete Moment Information [article]

Didier Henrion
2021 arXiv   pre-print
The resulting sequence of optimal solutions converges to the whole moment sequence of the measure which is shown to be the unique optimal solution of a certain infinite-dimensional linear optimization  ...  We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as for instance in some problems of optimal transport  ...  As for transport problem, we show that the measure µ to recover is the unique optimal solution of a certain infinite-dimensional LP on a measure space. II.  ... 
arXiv:2011.05170v2 fatcat:6uuf7rszmzgvfmnwsuyfl53yri

Transform-based particle filtering for elliptic Bayesian inverse problems [article]

Sangeetika Ruchi and Svetlana Dubinkina and Marco Iglesias
2019 arXiv   pre-print
We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements.  ...  We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial  ...  Acknowledgments This work is part of the research programme Shell-NWO/FOM Computational Sciences for Energy Research (CSER) with project number 14CSER007 which is partly financed by the Netherlands Organization  ... 
arXiv:1901.04706v2 fatcat:culzmo7nu5ec3kbjtq4k2smstm
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