翻訳と辞書 Words near each other ・ Polynomial and rational function modeling ・ Polynomial arithmetic ・ Polynomial basis ・ Polynomial chaos ・ Polynomial code ・ Polynomial conjoint measurement ・ Polynomial decomposition ・ Polynomial delay ・ Polynomial Diophantine equation ・ Polynomial expansion ・ Polynomial function theorems for zeros ・ Polynomial greatest common divisor ・ Polynomial hierarchy ・ Polynomial identity ring ・ Polynomial interpolation ・ Polynomial kernel ・ Polynomial least squares ・ Polynomial lemniscate ・ Polynomial long division ・ Polynomial matrix ・ Polynomial regression ・ Polynomial remainder theorem ・ Polynomial representations of cyclic redundancy checks ・ Polynomial ring ・ Polynomial sequence ・ Polynomial signal processing ・ Polynomial SOS ・ Polynomial texture mapping ・ Polynomial transformations ・ Polynomial Wigner–Ville distribution Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Polynomial kernel ： ウィキペディア英語版 Polynomial kernel In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models. Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of regression analysis, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of polynomial regression, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to logical conjunctions of input features.〔Yoav Goldberg and Michael Elhadad (2008). splitSVM: Fast, Space-Efficient, non-Heuristic, Polynomial Kernel Computation for NLP Applications. Proc. ACL-08: HLT.〕 ==Definition== For degree- polynomials, the polynomial kernel is defined as〔http://www.cs.tufts.edu/~roni/Teaching/CLT/LN/lecture18.pdf〕 :$K(x,y)\; =\; (x^\backslash mathsf\; y\; +\; c)^$ where and are vectors in the ''input space'', i.e. vectors of features computed from training or test samples and is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial. When , the kernel is called homogeneous. (A further generalized polykernel divides by a user-specified scalar parameter .) As a kernel, corresponds to an inner product in a feature space based on some mapping : :$K(x,y)\; =\; \backslash langle\; \backslash varphi(x),\; \backslash varphi(y)\; \backslash rangle$ The nature of can be seen from an example. Let , so we get the special case of the quadratic kernel. After using the multinomial theorem (twice—the outermost application is the binomial theorem) and regrouping, :$K(x,y)\; =\; \backslash left(\backslash sum\_^n\; x\_i\; y\_i\; +\; c\backslash right)^2\; =$ \sum_^n \left(x_i^2\right) \left(y_i^2 \right) + \sum_^n \sum_^ \left( \sqrt x_i x_j \right) \left( \sqrt y_i y_j \right) + \sum_^n \left( \sqrt x_i \right) \left( \sqrt y_i \right) + c^2 From this it follows that the feature map is given by: :$$ \varphi(x) = \langle x_n^2, \ldots, x_1^2, \sqrt x_n x_, \ldots, \sqrt x_n x_1, \sqrt x_ x_, \ldots, \sqrt x_ x_, \ldots, \sqrt x_ x_, \sqrt x_n, \ldots, \sqrt x_1, c \rangle 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「Polynomial 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.