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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 submodular set function ： ウィキペディア英語版 submodular set function In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function, that a single element makes when added to an input set, decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains.〔〔〔〔 == Definition == If $\backslash Omega$ is a finite set, a submodular function is a set function $f:2^\backslash rightarrow\; \backslash mathbb$, where $2^\backslash Omega$ denotes the power set of $\backslash Omega$, which satisfies one of the following equivalent definitions. # For every $X,\; Y\; \backslash subseteq\; \backslash Omega$ with $X\; \backslash subseteq\; Y$ and every $x\; \backslash in\; \backslash Omega\; \backslash backslash\; Y$ we have that $f(X\backslash cup\; \backslash )-f(X)\backslash geq\; f(Y\backslash cup\; \backslash )-f(Y)$. # For every $S,\; T\; \backslash subseteq\; \backslash Omega$ we have that $f(S)+f(T)\backslash geq\; f(S\backslash cup\; T)+f(S\backslash cap\; T)$. # For every $X\backslash subseteq\; \backslash Omega$ and $x\_1,x\_2\backslash in\; \backslash Omega\backslash backslash\; X$ we have that $f(X\backslash cup\; \backslash )+f(X\backslash cup\; \backslash )\backslash geq\; f(X\backslash cup\; \backslash )+f(X)$. A nonnegative submodular function is also a subadditive function, but a subadditive function need not be submodular. If $\backslash Omega$ is not assumed finite, then the above conditions are not equivalent. In particular a function $f$ defined by $f(S)\; =\; 1$ if $S$ is finite and $f(S)\; =\; 0$ if $S$ is infinite satisfies the first condition above, but the second condition fails when $S$ and $T$ are infinite sets with finite intersection. 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「submodular set function」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.