An Infinite Dimensional Analysis of Kernel Principal Components

2019/06/15 by Palle E. T. Jørgensen, Jorgensen, Palle E. T., Sooran Kang +5
Computer Science · Engineering · Mathematics · #34L16 #37M25 #42C15 #42C40 #47A70 #47B06 #47B32 #62H25 #65D15 #65K10 #65T60 #FOS: Mathematics #Functional Analysis (math.FA) #Image Processing Techniques and Applications #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1906.06451

openalex publication_date 2019/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study non-linear data-dimension reduction. We are motivated by the classical linear framework of Principal Component Analysis. In nonlinear case, we introduce instead a new kernel-Principal Component Analysis, manifold and feature space transforms. Our results extend earlier work for probabilistic Karhunen-Loève transforms on compression of wavelet images. Our object is algorithms for optimization, selection of efficient bases, or components, which serve to minimize entropy and error; and hence to improve digital representation of images, and hence of optimal storage, and transmission. We prove several new theorems for data-dimension reduction. Moreover, with the use of frames in Hilbert space, and a new Hilbert-Schmidt analysis, we identify when a choice of Gaussian kernel is optimal.

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