Learning and approximation by Gaussians on Riemannian manifolds

Learning function relations or understanding structures of data lying in manifolds embedded in huge dimensional Euclidean spaces is an important topic in learning theory. In this paper we study the approximation and learning by Gaussians of functions defined on a d-dimensional connected compact C Riemannian submanifold of IRn which is isometrically embedded. We show that the convolution with the Gaussian kernel with variance provides the uniform approximation order of O(s) when the approximated function is Lipschitz s (0, 1]. The uniform normal neighborhoods of a compact Riemannian manifold play a central role in deriving the approximation order. This approximation result is used to investigate the regression learning algorithm generated by the multi-kernel least square regularization scheme associated with Supported partially by the Research Grants Council of Hong Kong [Project No. CityU 103405], City University of Hong Kong [Project No. 7001983], National Science Fund for Distingu...

• Type: Journal
• Year: 2008
• Where: ADCM
• Authors: Gui-Bo Ye, Ding-Xuan Zhou
• Keywords: ADCM 2008, Approximation Order, Dimensional Euclidean Spaces, Uniform Approximation Order