Regularized principal component analysis

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• 作者：Yonathan Aflalo ; Ron Kimmel
• 关键词：Laplace ; Beltrami operator ; Principal component analysis ; Isometry
• 刊名：Chinese Annals of Mathematics, Series B
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：38
• 期：1
• 页码：1-12
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Applications of Mathematics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1860-6261
• 卷排序：38

文摘

Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis (PCA for short) approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for representation smooth functions sampled on regular grid. It is derived as the eigenfunctions of the circulant Laplacian operator. In this paper, based on the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO for short), the construction of PCA for geometric structures is regularized. By assuming smoothness of a given data, one could exploit the intrinsic geometric structure to regularize the construction of a basis by which the observed data is represented. The LBO can be decomposed to provide a representation space optimized for both internal structure and external observations. The proposed model takes the best from both the intrinsic and the extrinsic structures of the data and provides an optimal smooth representation of shapes and forms.