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We study non-linear data-dimension reduction. We are motivated by theclassical linear framework of Principal Component Analysis. In nonlinear case,we introduce instead a new kernel-Principal Component Analysis, manifold andfeature space transforms. Our results extend earlier work for probabilisticKarhunen-Lo\`eve transforms on compression of wavelet images. Our object isalgorithms for optimization, selection of efficient bases, or components, whichserve to minimize entropy and error; and hence to improve digitalrepresentation of images, and hence of optimal storage, and transmission. Weprove several new theorems for data-dimension reduction. Moreover, with the useof frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identifywhen a choice of Gaussian kernel is optimal.

An Infinite Dimensional Analysis of Kernel Principal Components