COMPARISON BETWEEN W_2 DISTANCE AND H~(-1) NORM, AND LOCALIZATION OF WASSERSTEIN DISTANCE

Peyre Remi

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COMPARISON BETWEEN W_2 DISTANCE AND H~(-1) NORM, AND LOCALIZATION OF WASSERSTEIN DISTANCE

机译：W_2距离与H〜（-1）范数的比较以及WASSERSTEIN距离的定位

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It is well known that the quadratic Wasserstein distance W-2(., .) is formally equivalent, for infinitesimally small perturbations, to some weighted H-1 homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the W-2 distance exhibits some localization phenomenon: if mu and nu are measures on R-n and phi: R-n - R+ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between phi . mu and phi . nu by an explicit multiple of W-2(mu, nu).

机译：众所周知，对于无限小的微小扰动，二次Wasserstein距离W-2（。，。）在形式上等同于某些加权H-1齐次Sobolev范数。在本文中，我证明了可以等价地综合这些距离之间的非渐近比较结果。然后，我用这些结果来证明W-2距离表现出某种局部化现象：如果mu和nu是Rn和phi的量度：Rn-> R +是具有紧实支持的一些碰撞函数，那么在温和假设下，您可以绑定到phi之间的Wasserstein距离之上。亩和.。 nu乘以W-2（mu，nu）的倍数。

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来源
《ESAIM》 |2018年第4期|1489-1501|共13页
作者
Peyre Remi;
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作者单位

Inst Elie Cartan Lorraine, Campus Aiguillettes,1 Blvd Aiguillettes,BP 70239, F-54506 Vandoeuvre Les Nancy, France;

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原文格式 PDF
正文语种 eng
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关键词
Wasserstein distance; homogeneous Sobolev norm; localization;

机译：Wasserstein距离;齐次Sobolev范数;局部化;

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2. Analysis on comparison of distances derived by one-norm and two-norm with weight functions [J] . Deng P.S., Chao H.C.J. Applied mathematics and computation . 2013,第17期

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3. Entropic approach to e. rio's central limit theorem for w_2 transport distance [J] . Bobkov S.G. Statistics & Probability Letters . 2013,第7期

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COMPARISON BETWEEN W_2 DISTANCE AND H~(-1) NORM, AND LOCALIZATION OF WASSERSTEIN DISTANCE

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