Wasserstein barycenters over Riemannian manifolds

Kim, Young-Heon; Pass, Brendan

doi:10.1016/j.aim.2016.11.026

Cited by 54 publications

(67 citation statements)

References 42 publications

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“…Notice that if f : Ω Ñ R is real-valued and harmonic, then for any ε ą 0 f pξq is the barycenter of f pηq for η P Bpξ, εq, while in the metric case this property only holds asymptotically as ε Ñ 0. For barycenters in the Wasserstein space, there exists a generalized Jensen inequality: it was already proved in the finite case by Agueh and Carlier [1,Proposition 7.6] under the assumption that F is convex along generalized geodesics, and in a more general case (in particular with an infinite numbers of measures defined on a compact manifold, whereas Agueh and Carlier worked in the Euclidan space) by Kim and Pass [21,Section 7], but with rather strong regularity assumptions on the measures. We reprove this Jensen inequality in a case adapted to our context by letting the barycenter µ ε pξq follow the gradient of the functional F (for gradients flows in the Wasserstein space see [4]) and use the result as a competitor: through arguments first advanced in [25] in a very different context under the name of flow interchange, one can show (estimating the derivative of the Wasserstein distance along the flow of F with the so-called (EVI) inequality) that for a.e.…”

Section: Main Definitions and Resultsmentioning

confidence: 99%

Harmonic mappings valued in the Wasserstein space

Lavenant

2019

Journal of Functional Analysis

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We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings µ : Ω Ñ pPpDq, W2q defined over a subset Ω of R p and valued in the space PpDq of probability measures on a compact convex subset D of R q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Korevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces.We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary BΩ are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Ω a segment of R. As the Wasserstein space pPpDq, W2q is positively curved in the sense of Alexandrov we cannot apply the theory of Korevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on BΩ, uniqueness is an open question.If Ω is a segment of R, it is known that a curve valued in the Wasserstein space PpDq can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Ω Ñ PpDq cannot be represented as the superposition of mappings Ω Ñ D.We are able to show the validity of a maximum principle: the composition F˝µ of a function F : PpDq Ñ R convex along generalized geodesics and a harmonic mapping µ : Ω Ñ PpDq is a subharmonic real-valued function.We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of pPpDq, W2q which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport. 14 3. The Dirichlet energy and the space H 1 pΩ, PpDqq 15 3.1. A Benamou-Brenier type definition 17 3.2. The smooth case 20

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Section: Main Definitions and Resultsmentioning

confidence: 99%

Harmonic mappings valued in the Wasserstein space

Lavenant

2019

Journal of Functional Analysis

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“…The general study of barycenters in the Wasserstein space was initiated by Agueh-Carlier [1] when the underlying space X = R n is Euclidean, and continued by the present authors to the setting where X is a smooth Riemannian manifold [8]. These represent a natural, non-linear way to interpolate between several (or infinitely many) probability measures, and have received a great deal of attention in recent years, due in part to important applications in image processing and statistics; see, for example, the work of Rabin et al [11] and Bigot and Klein [2] among others.…”

Section: Introductionmentioning

confidence: 96%

A Canonical Barycenter via Wasserstein Regularization

Kim¹,

Pass²

2018

SIAM J. Math. Anal.

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We introduce a weak notion of barycenter of a probability measure µ on a metric measure space (X, d, m), with the metric d and reference measure m. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter B(µ) is well defined; it is a probability measure on X supported on the set of the usual metric barycenter points of the given measure µ. The definition uses the canonical embedding of the metric space X into its Wasserstein space P (X), pushing a given measure µ forward to a measure on P (X). We then regularize the measure by the Wasserstein distance to the reference measure m, and obtain a uniquely defined measure on X supported on the barycentric points of µ. We investigate various properties of B(µ).

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“…The Problem 3 is to minimise K among all ρ ∈ P 2 . It coïncides with the notion of Wasserstein Barycenters, see [1], [14], when…”

Section: 32mentioning

confidence: 98%