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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 Proximal gradient method ： ウィキペディア英語版 Proximal gradient method Proximal gradient methods are a generalized form of projection used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of form :$$ \operatorname_ \qquad f_1(x) +f_2(x) + \cdots+ f_(x) +f_n(x) where $f\_1,\; f\_2,\; ...,\; f\_n$ are convex functions defined from $f:\; \backslash mathbb^N\; \backslash rightarrow\; \backslash mathbb$ where some of the functions are non-differentiable, this rules out our conventional smooth optimization techniques like Steepest descent method, conjugate gradient method etc. There is a specific class of algorithms which can solve above optimization problem. These methods proceed by splitting, in that the functions $f\_1,\; .\; .\; .\; ,\; f\_n$ are used individually so as to yield an easily implementable algorithm. They are called proximal because each non smooth function among $f\_1,\; .\; .\; .\; ,\; f\_n$ is involved via its proximity operator. Iterative Shrinkage thresholding algorithm, projected Landweber, projected gradient, alternating projections, alternating-direction method of multipliers, alternating split Bregman are special instances of proximal algorithms. Details of proximal methods are discussed in Combettes and Pesquet.〔 〕 For the theory of proximal gradient methods from the perspective of and with applications to statistical learning theory, see proximal gradient methods for learning. == Notations and terminology == Let $\backslash mathbb^N$, the $N$-dimensional euclidean space, be the domain of the function $f:\; \backslash mathbb^N\; \backslash rightarrow\; (-\backslash infty,+\backslash infty]$. Suppose $C$ is a non-empty convex subset of $\backslash mathbb^N$. Then, the indicator function of $C$ is defined as : $i\_C\; :\; x\; \backslash mapsto$ \begin 0 & \text x \in C \\ + \infty & \text x \notin C \end : $p$-norm is defined as ( $\backslash |\; \backslash cdot\; \backslash |\_p$ ) : $$ \|x\|_p = ( |x_1|^p + |x_2|^p + \cdots + |x_N|^p )^ The distance from $x\; \backslash in\; \backslash mathbb^N$ to $C$ is defined as : $$ D_C(x) = \min_ \|x - y\| If $C$ is closed and convex, the projection of $x\; \backslash in\; \backslash mathbb^N$ onto $C$ is the unique point $P\_Cx\; \backslash in\; C$ such that $D\_C(x)\; =\; \backslash |\; x\; -\; P\_Cx\; \backslash |\_2$. The subdifferential of $f$ is given by : $$ \partial f = \^N, (y-x)^\mathrmu+f(x) \leq f(y)).\} 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「Proximal gradient method」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.