Palle E.T. Jorgensen , Myung-Sin Song , James Tian
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Abstract
With view to applications, we present here general classes of non-orthogonal expansions in Hilbert space. The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our work includes a diverse list of optimization problems, new Karhunen–Loève transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. For this we also make use of specific reproducing kernel Hilbert spaces, kernel factorizations, and finite-dimensional approximations. Our projection algorithms are designed with view to maximum likelihood solutions, minimization of “cost” problems, identification of principal components, and data-dimension reduction.
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