The Monge problem in Rd

Champion, Thierry; Pascale, Luigi De

doi:10.1215/00127094-1272939

Cited by 61 publications

(81 citation statements)

References 20 publications

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“…Later, existence of optimal maps for the case c(x, y) := x − y , · being any norm has been established, at increasing levels of generality, in [9], [28], [27] (containing the most general result, for any norm) and [25].…”

Section: Bibliographical Notesmentioning

confidence: 99%

A User’s Guide to Optimal Transport

Ambrosio

Gigli

2012

Lecture Notes in Mathematics

303

405

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This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.

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Section: Bibliographical Notesmentioning

confidence: 99%

A User’s Guide to Optimal Transport

Ambrosio

Gigli

2012

Lecture Notes in Mathematics

303

405

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“…Of particular interest is the result of [15], later extended by [55], which shows that when c(x, T (x)) is quadratic and µ ref is atomless, the optimal transport map exists and is unique; moreover this map is the gradient of a convex function and thus is monotone. Generalizations of this result accounting for different cost functions and spaces can be found in [19,2,27,11]. For a thorough contemporary development of optimal transport we refer to [88,87].…”

Section: Transport Maps and Optimal Transportmentioning

confidence: 96%

“…The choice of direction in (19) involves integration over the target distribution, as in the objective function of (20), which we approximate using the given samples. Let (…”

Section: S ∈ Tmentioning

confidence: 99%

Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

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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. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available, and second, when the target distribution is known only through a finite collection of samples. We show that in both settings the desired transports can be characterized as the solutions of variational problems. 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 of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to "Gaussianize" complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.

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“…Optimal transport problems is by now a classical subject that still deserves attention. We refer to [2], [3], [4], [6], [21], [22], [23] and the surveys and books [1], [13], [25] and [26]. It has many applications, for example in economics (matching problems), [5], [7], [8], [9], [10], [11], [20].…”

Section: 2mentioning

confidence: 99%