# 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

arXiv: https://arxiv.org/abs/2601.18080 · HTML: https://mlsanity.com/paper/2601.18080/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 55 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Builds on (most cited first, 10 of 10)

- [Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm](https://mlsanity.com/paper/1310.5715/) (2013)
- [An Accelerated Randomized Kaczmarz Algorithm](https://mlsanity.com/paper/1310.2887/) (2013)
- [Convex Formulation for Kernel PCA and its Use in Semi-Supervised Learning](https://mlsanity.com/paper/1610.06811/) (2016)
- [Bayesian model and dimension reduction for uncertainty propagation: applications in random media](https://mlsanity.com/paper/1711.02475/) (2017)
- [Smooth Sensitivity Based Approach for Differentially Private Principal Component Analysis](https://mlsanity.com/paper/1710.10556/) (2017)
- [Reproducing kernel Hilbert spaces in the mean field limit](https://mlsanity.com/paper/2302.14446/) (2023)
- [A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS L2 spaces](https://mlsanity.com/paper/1904.04414/) (2019)
- [A Weighted Randomized Kaczmarz Method for Solving Linear Systems](https://mlsanity.com/paper/2007.02910/) (2020)
- [Operator theory, kernels, and Feedforward Neural Networks](https://mlsanity.com/paper/2301.01327/) (2023)
- [Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and λ-Effectiveness](https://mlsanity.com/paper/2511.07604/) (2025)

## Built on by (most cited first, 1 of 1)

- [The role of parameter Jacobians in the stability of network outputs](https://mlsanity.com/paper/2608.27748/) (2026)

## Abstract (arXiv)

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 λn-relaxed variants, and λn-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.