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SVM LEARNING AND L-p APPROXIMATION BY GAUSSIANS ON RIEMANNIAN MANIFOLDS

机译：高斯人在黎曼流形上的SVM学习和L-p逼近

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摘要

We confirm by the multi-Gaussian support vector machine (SVM) classification that the information of the intrinsic dimension of Riemannian manifolds can be used to illustrate the efficiency (learning rates) of learning algorithms. We study an approximation scheme realized by convolution operators involving the Gaussian kernels with flexible variances. The essential analysis lies in the study of its approximation order in L-p (1 <= p < infinity) norm as the variance of the Gaussian tends to zero. It is different from the analysis for approximation in C(X) since pointwise estimations do not work any more. The L-p approximation arises from the SVM case where the approximated function is the Bayes rule and is not continuous, in general. The approximation error is estimated by imposing a regularity condition that the approximated function lies in some interpolation spaces. Then, the learning rates for multi-Gaussian regularized classifiers with general classification loss functions are derived, and the rates depend on the intrinsic dimension of the Riemannian manifold instead of the dimension of the underlying Euclidean space. Here, the input space is assumed to be a connected compact C-infinity Riemannian submanifold of R-n. The uniform normal neighborhoods of the Riemannian manifold and the radial basis form of Gaussian kernels play an important role.
机译：我们通过多高斯支持向量机（SVM）分类确认，黎曼流形内在维数的信息可用于说明学习算法的效率（学习率）。我们研究了卷积算子实现的一种近似方案，该卷积算子涉及具有高阶方差的高斯核。本质分析在于，当高斯方差趋于零时，研究其在L-p（1 <= p <无穷大）范数中的近似阶数。它与C（X）中的逼近分析不同，因为逐点估算不再起作用。 L-p近似值是从SVM情况得出的，其中，近似函数是贝叶斯规则，通常不连续。通过施加规则性条件来估计近似误差，该规则性条件是近似函数位于某些插值空间中。然后，推导具有一般分类损失函数的多高斯正则分类器的学习率，该学习率取决于黎曼流形的内在维数而不是基础欧几里德空间的维数。在此，输入空间被假定为R-n的连通紧致C无限黎曼子流形。黎曼流形的统一法线邻域和高斯核的径向基础形式起着重要作用。

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