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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 Radial basis function kernel ： ウィキペディア英語版 Radial basis function kernel In machine learning, the (Gaussian) radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.〔Yin-Wen Chang, Cho-Jui Hsieh, Kai-Wei Chang, Michael Ringgaard and Chih-Jen Lin (2010). ("Training and testing low-degree polynomial data mappings via linear SVM" ). ''J. Machine Learning Research'' 11:1471–1490.〕 The RBF kernel on two samples x and x', represented as feature vectors in some ''input space'', is defined as〔Vert, Jean-Philippe, Koji Tsuda, and Bernhard Schölkopf (2004). ("A primer on kernel methods". ) ''Kernel Methods in Computational Biology''.〕 :$K(\backslash mathbf,\; \backslash mathbf)\; =\; \backslash exp\backslash left(-\backslash frac||^2\}\backslash right)$ $\backslash textstyle||\backslash mathbf\; -\; \backslash mathbf||^2$ may be recognized as the squared Euclidean distance between the two feature vectors. $\backslash sigma$ is a free parameter. An equivalent, but simpler, definition involves a parameter $\backslash textstyle\backslash gamma\; =\; \backslash tfrac$: :$K(\backslash mathbf,\; \backslash mathbf)\; =\; \backslash exp(-\backslash gamma||\backslash mathbf\; -\; \backslash mathbf||^2)$ Since the value of the RBF kernel decreases with distance and ranges between zero (in the limit) and one (when ), it has a ready interpretation as a similarity measure.〔 The feature space of the kernel has an infinite number of dimensions; for $\backslash sigma\; =\; 1$, its expansion is: :$\backslash exp\backslash left(-\backslash frac||\backslash mathbf\; -\; \backslash mathbf||^2\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash frac)^j\}\; \backslash exp\backslash left(-\backslash frac||\backslash mathbf||^2\backslash right)$ \exp\left(-\frac||\mathbf||^2\right) ==Approximations== Because support vector machines and other models employing the kernel trick do not scale well to large numbers of training samples or large numbers of features in the input space, several approximations to the RBF kernel (and similar kernels) have been devised.〔Andreas Müller (2012). (Kernel Approximations for Efficient SVMs (and other feature extraction methods) ).〕 Typically, these take the form of a function ''z'' that maps a single vector to a vector of higher dimensionality, approximating the kernel: :$z(\backslash mathbf)z(\backslash mathbf)\; \backslash approx\; \backslash varphi(\backslash mathbf)\backslash varphi(\backslash mathbf)\; =\; K(\backslash mathbf,\; \backslash mathbf)$ where $\backslash textstyle\backslash varphi$ is the implicit mapping embedded in the RBF kernel. One way to construct such a ''z'' is to randomly sample from the Fourier transformation of the kernel.〔Ali Rahimi and Benjamin Recht (2007). ("Random features for large-scale kernel machines" ). ''Neural Information Processing Systems''.〕 Another approach uses the Nyström method to approximate the eigendecomposition of the Gram matrix ''K'', using only a random sample of the training set. 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「Radial basis function kernel」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.