Positive Definite Kernels, Algorithms, Frames, and Approximations

TL;DR

This paper introduces a novel recursive projection algorithm framework in Hilbert spaces for optimization, PCA, and kernel methods, leveraging reproducing kernel Hilbert spaces and finite-dimensional approximations.

Contribution

It presents a new approach to designing Kaczmarz-type algorithms using recursive projection systems for diverse optimization and data analysis tasks.

Findings

Effective PCA probability computations

Enhanced variance data analysis

Improved kernel-based approximations

Abstract

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 applications include a diverse list of optimization problems, new Karhunen-Lo\`eve transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. Specifically, we apply our recursive projection algorithms for new computations of PCA probabilities and of variance data. For this we also make use of specific reproducing kernel Hilbert spaces, factorization for kernels, 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.