Existence and consistency of Wasserstein barycenters

Gouic, Thibaut Le; Loubes, Jean–Michel

doi:10.1007/s00440-016-0727-z

Cited by 75 publications

(101 citation statements)

References 23 publications

(37 reference statements)

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“…The inner minimization problem is well posed and admits a unique solution

μ_{frakturM} (w)

for any

p \in (1, \infty)

and any w ∈Δ N − 1 . This follows from the general results of ( p = 2) and , , but for the sake of completeness, we provide the following argument which may be used as an alternative proof of the existence of the p ‐Wasserstein barycenter, which we think is of interest in its own right. Define the functional

q \mapsto J (q, w) : = \sum_{i = 1}^{N} w_{i} \int_{0}^{1} | q (s) - q_{i} (s) |^{p} ds

and consider it as a functional

J : scriptS \to {double-struckR}_{+}

, where

scriptS \subset L^{p} (0, 1)

is the set of quantile functions (importantly this is not a vector space).…”

Section: Well‐posedeness Results Of Learning Schemes Based On the Wasmentioning

confidence: 78%

“…The inner minimization problem (1) is well posed and admits a unique solution M .w/ for any p 2 .1, 1/ and any w 2 N 1 . This follows from the general results of [4] (p D 2) and [5], [6], but for the sake of completeness, we provide the following argument which may be used as an alternative proof of the existence of the p-Wasserstein barycenter, which we think is of interest in its own right. Define the functional q 7 !…”

Section: Proofmentioning

confidence: 83%

“…Choosing ı i small enough, the second contribution may also be made smaller than =2. Hence J enjoys property (6). C. The value function w 7 !…”

Section: Proofmentioning

confidence: 99%

“…w. Fix > 0. By (6), there exists ı > 0 such that for every w 0 with the property kw w 0 k < ı, there exists q 0 2 L p .0, 1/ such that J.q 0 , w 0 / < J.q, w/C . For this ı, and because w k !…”

Section: Proofmentioning

confidence: 99%

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A learning algorithm for source aggregation

Papayiannis

Yannacopoulos

2016

Math Methods in App Sciences

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The problem of model aggregation from various information sources of unknown validity is addressed in terms of a variational problem in the space of probability measures. A weight allocation scheme to the various sources is proposed, which is designed to lead to the best aggregate model compatible with the available data and the set of prior measures provided by the information sources. Copyright © 2016 John Wiley & Sons, Ltd.

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“…The inner minimization problem is well posed and admits a unique solution

μ_{frakturM} (w)

for any

p \in (1, \infty)

q \mapsto J (q, w) : = \sum_{i = 1}^{N} w_{i} \int_{0}^{1} | q (s) - q_{i} (s) |^{p} ds

and consider it as a functional

J : scriptS \to {double-struckR}_{+}

, where

scriptS \subset L^{p} (0, 1)

is the set of quantile functions (importantly this is not a vector space).…”

Section: Well‐posedeness Results Of Learning Schemes Based On the Wasmentioning

confidence: 78%

Section: Proofmentioning

confidence: 83%

“…Choosing ı i small enough, the second contribution may also be made smaller than =2. Hence J enjoys property (6). C. The value function w 7 !…”

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

See 2 more Smart Citations

A learning algorithm for source aggregation

Papayiannis

Yannacopoulos

2016

Math Methods in App Sciences

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“…As points on this space corresponds to probability measures, the Wasserstein barycenter of a collection of probability measures

scriptM

corresponds to the best approximation of the entire collection by a single probability measure. Wasserstein barycenters are a very active field of research recently and have been studied from various perspectives (see, e.g., Agueh & Carlier, ; Kim & Pass, ; Le Gouic & Loubes, ) including statistical learning (Papayiannis & Yannacopoulos, ).…”

Section: Weight Selection Approaches For the Aggregate Modelsmentioning

confidence: 99%

Model aggregation using optimal transport and applications in wind speed forecasting

2018

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In several environmental affected socio‐economic activities, including renewable energy site assessment, search and rescue operations, and local microclimate modeling, the need of very local wind speed prediction is critical and not completely covered by the use of numerical weather prediction models. In meteorology and, particularly, in wind speed prediction, the spatial location of the prediction does not coincide with the spatial locations where numerical models provide estimates of the relevant quantity (which are typically grid points used for the numerical resolution of the wind transport equations). Hence, the important problem of constructing a predictive model for the wind speed at the required location using a combination of actual measurements and model predictions arises. This problem is far from trivial on account of the fact that measurements and predictions do not refer to the same quantity for the reason that typical grid points for the numerical scheme that provide model predictions and the location of the meteorological stations that provide measurements do not coincide. In this work, a new approach is proposed based on optimal transportation theory for the aggregation of model predictions and measurements for the construction of an optimal predictor for wind speed at the location of interest. Our model provides a linear predictive model in the space of probability distributions of the predictors (Wasserstein space), which is then mapped into observation space using a generalized quantile regression technique. Importantly, the proposed scheme allows also for the construction of zone monitoring the extremes, which when applied to real data, provides superior results with respect to other existing methods.

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Gradient Methods for Problems with Inexact Model of the Objective

Stonyakin

Dvinskikh

Dvurechensky

et al. 2019

Mathematical Optimization Theory and Operations Research

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We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes inexact oracle [19] and relative smoothness condition [43]. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [49]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments.

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Existence and consistency of Wasserstein barycenters

Cited by 75 publications

References 23 publications

A learning algorithm for source aggregation

A learning algorithm for source aggregation

Model aggregation using optimal transport and applications in wind speed forecasting

Gradient Methods for Problems with Inexact Model of the Objective

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