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

TL;DR

This paper introduces a novel operator theoretic framework for analyzing complex systems and machine learning algorithms, providing new bounds, convergence results, and stability criteria for residuals and kernel methods.

Contribution

It develops a new mathematical framework using operator identities to analyze residuals in complex systems and machine learning, with explicit convergence and stability results.

Findings

Proves new bounds on energy residuals in complex systems.

Provides convergence results for greedy Kernel PCA algorithms.

Establishes stability criteria for kernel-based methods under noise.

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 $\lambda_{n}$-relaxed variants, and $\lambda_{n}$-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.