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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, James F. Tian

TL;DR

This work develops an operator-theoretic framework for multi-channel systems with intrinsic subdivisions, using a sequence of relaxations $\{\lambda_n\}$ to derive a telescoping energy-residual identity and a priori energy bounds. By embedding the theory in RKHS and introducing tree-based kernel reparametrization and greedy refinement, it yields exact energy decompositions, convergence guarantees, and stability under noise for kernel interpolation and greedy kernel PCA. Key contributions include a general telescoping identity $I-T_N^*T_N$, a summability condition ensuring effectiveness, and practical methods for kernel compression with rigorous error control via discarded energy. The results offer a principled, online-friendly alternative to classical kernel methods, with explicit energy accounting, scalability through multichannel splitting, and robust performance under stochastic sampling and 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 $λ_{n}$-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.

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 work develops an operator-theoretic framework for multi-channel systems with intrinsic subdivisions, using a sequence of relaxations to derive a telescoping energy-residual identity and a priori energy bounds. By embedding the theory in RKHS and introducing tree-based kernel reparametrization and greedy refinement, it yields exact energy decompositions, convergence guarantees, and stability under noise for kernel interpolation and greedy kernel PCA. Key contributions include a general telescoping identity , a summability condition ensuring effectiveness, and practical methods for kernel compression with rigorous error control via discarded energy. The results offer a principled, online-friendly alternative to classical kernel methods, with explicit energy accounting, scalability through multichannel splitting, and robust performance under stochastic sampling and 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 -relaxed variants, and -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.
Paper Structure (11 sections, 24 theorems, 193 equations)

This paper contains 11 sections, 24 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Multi-channel defect splitting
  3. Defect decomposition
  4. Kaczmarz-$\lambda P$ schemes
  5. Telescoping, summability criteria, and effectiveness
  6. Multichannel tree splitting
  7. Kernel interpolation
  8. Random Dropouts (stochastic sampling)
  9. Kernel reparametrization from tree-splitting
  10. Greedy refinement and kernel compression
  11. Connections to Greedy Kernal Principal Component Analysis

Key Result

Proposition 2.1

Let $A_{1},A_{2},\dots$ be contractions on $H$. Set $T_{0}=I$ and $T_{n}=A_{n}A_{n-1}\cdots A_{1}$ for $n\ge1$. Then for every $x\in H$ and every $N\ge1$, Equivalently, for every $N\ge1$, where the sum is finite and hence unambiguous in $B\left(H\right)$.

Theorems & Definitions (63)

  • Proposition 2.1
  • proof
  • Definition 2.2: Effectiveness/admissibility
  • Example 2.3: jeong2025
  • Theorem 2.4
  • proof
  • Remark : Why variable relaxation can matter
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 53 more