Generalization Error Analysis for Polynomial Kernel Methods - Algebraic Geometrical Approach
Generalization Error Analysis for Polynomial Kernel Methods - Algebraic Geometrical Approach
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多项式核方法的泛化误差分析 - 代数几何方法
DOI：
10.1007/3-540-44989-2_25
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
K. Ikeda
中科院分区：
文献类型：
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作者：
K. Ikeda
本文研究了具有多项式核函数的学习分类器的泛化性能。我们首先证明了学习机的泛化误差取决于分离曲线的性质，即输入曲面与特征空间中真实分离超平面的交点。当输入空间是一维时，问题被分解为与相交点的数目一样多的一维问题。否则，推广误差由分离曲线的类别决定。接下来，我们考虑分离曲线的类如何取决于真实的分离函数。当真分离多项式函数不可约时，类是最大的，否则更小。在这两种情况下，类只依赖于真函数，而不依赖于特征空间的维数。结果表明，即使当特征空间的维数变大时，泛化误差也不会增加，并且在核学习中不会发生所谓的过建模。
The generalization properties of learning classifiers with a polynomial kernel function are examined here. We first show that the generalization error of the learning machine depends on the properties of the separating curve, that is, the intersection of the input surface and the true separating hyperplane in the feature space. When the input space is one-dimensional, the problem is decomposed to as many one-dimensional problems as the number of the intersecting points. Otherwise, the generalization error is determined by the class of the separating curve. Next, we consider how the class of the separating curve depends on the true separating function. The class is maximum when the true separating polynomial function is irreducible and smaller otherwise. In either case, the class depends only on the true function and does not on the dimension of the feature space. The results imply that the generalization error does not increase even when the dimension of the feature space gets larger and that the so-called overmodeling does not occur in the kernel learning.
核方法的几何和学习曲线 具有多项式核的多项式核
DOI：
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发表时间：
2004
期刊：
Systems and Computers in Japan 35巻7号
影响因子：
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作者：
Kazushi IKEDA;Noboru MURATA;Kazushi IKEDA;Kazushi IKEDA
通讯作者：
Kazushi IKEDA