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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Wasserstein metric ： ウィキペディア英語版 Wasserstein metric In mathematics, the Wasserstein (or Vasershtein) metric is a distance function defined between probability distributions on a given metric space ''M''. Intuitively, if each distribution is viewed as a unit amount of "dirt" piled on ''M'', the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of dirt that needs to be moved times the distance it has to be moved. Because of this analogy, the metric is known in computer science as the earth mover's distance. The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970, after the Russian mathematician Leonid Vaseršteĭn who introduced the concept in 1969. Most English-language publications use the German spelling "Wasserstein" (attributed to the name "Vasershtein" being of German origin). ==Definition== Let (''M'', ''d'') be a metric space for which every probability measure on ''M'' is a Radon measure (a so-called Radon space). For ''p'' ≥ 1, let ''P''''p''(''M'') denote the collection of all probability measures ''μ'' on ''M'' with finite ''p''th moment: for some ''x''0 in ''M'', : Then the ''p''th Wasserstein distance between two probability measures ''μ'' and ''ν'' in ''P''''p''(''M'') is defined as :$W\_\; (\backslash mu,\; \backslash nu):=\backslash left(\; \backslash inf\_\; \backslash int\_\; d(x,\; y)^\; \backslash ,\; \backslash mathrm\; \backslash gamma\; (x,\; y)\; \backslash right)^,$ where Γ(''μ'', ''ν'') denotes the collection of all measures on ''M'' × ''M'' with marginals ''μ'' and ''ν'' on the first and second factors respectively. (The set Γ(''μ'', ''ν'') is also called the set of all couplings of ''μ'' and ''ν''.) The above distance is usually denoted ''W''''p''(''μ'', ''ν'') (typically among authors who prefer the "Wasserstein" spelling) or ℓ''p''(''μ'', ''ν'') (typically among authors who prefer the "Vasershtein" spelling). The remainder of this article will use the ''W''''p'' notation. The Wasserstein metric may be equivalently defined by :$W\_\; (\backslash mu,\; \backslash nu)^\; =\; \backslash inf\; \backslash mathbf\; \backslash big(d(\; X\; ,\; Y\; )^\; \backslash big),$ where E() denotes the expected value of a random variable ''Z'' and the infimum is taken over all joint distributions of the random variables ''X'' and ''Y'' with marginals ''μ'' and ''ν'' respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wasserstein metric」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.