Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning

Palle E. T Jorgensen; Myung-Sin Song; James F Tian

doi:10.48550/arxiv.2601.18080

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Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning

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Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning

Palle E. T Jorgensen, Myung-Sin Song and James F Tian

ArXiv.org

Cornell University

01/26/2026

DOI: 10.48550/arxiv.2601.18080

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Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

We present a new operator theoretic framework for analysis of complex systems with intrinsic subdivisions into components, taking the form of "residuals" in general, and "telescoping energy residuals" in particular. We prove new results which yield admissibility/effectiveness, and new a priori bounds on energy residuals. Applications include infinite-dimensional Kaczmarz theory for$λ_{n}$ -relaxed variants, and$λ_{n}$ -effectiveness. And we give applications of our framework to generalized machine learning algorithms, greedy Kernel Principal Component Analysis (KPCA), proving explicit convergence results, residual energy decomposition, and criteria for stability under noise.

Computer Science - Learning

Mathematics - Functional Analysis

Mathematics - Operator Algebras

Details

Title: Subtitle: Use of operator defect identities in multi-channel signal plus residual-analysis via iterated products and telescoping energy-residuals: Applications to kernels in machine learning
Creators: Palle E. T Jorgensen
Myung-Sin Song
James F Tian
Resource Type: Preprint
Publication Details: ArXiv.org
DOI: 10.48550/arxiv.2601.18080
ISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 01/26/2026
Academic Unit: Mathematics
Record Identifier: 9985132181302771

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