Flexible Hypersurface Fitting with RBF Kernels-猫眼课题宝

DOI：

10.1007/978-3-642-40261-6_34

发表时间：

2013-08

期刊：

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中科院分区：

其他

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作者：

Jun Fujiki;S. Akaho

给出了一种基于RBF核函数的柔性超曲面拟合方法。为了将超曲面拟合到欧几里德空间中给定的点集合，我们可以将超平面拟合方法应用于映射到高维特征空间的点。这种拟合相当于通过消除特征空间中数据点的方差协方差矩阵的最小特征值所对应的特征向量所张成的线性空间，从而对特征空间进行一维约简。这种降维称为次要成分分析（MCA），它解决与核主成分分析相同的特征值问题，并提取最小特征值对应的特征向量。通常，特征空间被设定为欧几里得空间，即有限的希尔伯特空间。为了考虑无限希尔伯特空间的MCA，需要构造一个核MCA (KMCA)，它导致了在再现核希尔伯特空间中的MCA。然而，表征定理并不适用于KMCA，因为对于无限的希尔伯特空间，在MCA中会出现无限数量的零特征值。那么，拟合解在无限希尔伯特空间中是不唯一确定的，与有限希尔伯特空间中存在唯一解相反。这种模糊拟合会产生拟合的不稳定性，对拟合是不利的，但可以实现柔性拟合。基于这种灵活性，提出了一种利用RBF核函数在无限Hilbert空间中实现柔性超曲面拟合的方法。虽然考虑了每个样本处由核函数定义的矩阵的一些特征向量，但在特定情况下的模拟结果中，我们有一个合理解的候选解。仿真结果表明，该方法的灵活性是有效的。

This paper gives a method of flexible hypersurface fitting with RBF kernel functions. In order to fit a hypersurface to a given set of points in an Euclidean space, we can apply the hyperplane fitting method to the points mapped to a high dimensional feature space. This fitting is equivalent to a one-dimensional reduction of the feature space by eliminating the linear space spanned by an eigenvector corresponding to the smallest eigenvalue of a variance covariance matrix of data points in the feature space. This dimension reduction is called minor component analysis (MCA), which solves the same eigenvalue problem as kernel principal component analysis and extracts the eigenvector corresponding to the least eigenvalue. In general, feature space is set to an Euclidean space, which is a finite Hilbert space. To consider an MCA for an infinite Hilbert space, a kernel MCA (KMCA), which leads to an MCA in reproducing kernel Hilbert space, should be constructed. However, the representer theorem does not hold for a KMCA since there are infinite numbers of zero-eigenvalues would appear in an MCA for the infinite Hilbert space. Then, the fitting solution is not determined uniquely in the infinite Hilbert space, contrary to there being a unique solution in a finite Hilbert space. This ambiguity of fitting seems disadvantageous because it derives instability in fitting, but it can realize flexible fitting. Based on this flexibility, this paper gives a hypersurface fitting method in the infinite Hilbert space with RBF kernel functions to realize flexible hypersurface fitting. Although some eigenvectors of the matrix defined from kernel function at each sample are considered, we have a candidate of a reasonable solution among the simulation result under a specific situation. It is seen that the flexibility of our method is still effective through simulations.

Flexible Hypersurface Fitting with RBF Kernels

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