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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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915755495260160 |
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| author | Jorgensen, Palle E. T. Song, Myung-Sin Tian, James F. |
| author_facet | Jorgensen, Palle E. T. Song, Myung-Sin Tian, James F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18080 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 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 Jorgensen, Palle E. T. Song, Myung-Sin Tian, James F. Functional Analysis Machine Learning Operator Algebras Primary 47N70, Secondary 37N40, 37L55, 49Q15, 46E22 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. |
| title | 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 |
| topic | Functional Analysis Machine Learning Operator Algebras Primary 47N70, Secondary 37N40, 37L55, 49Q15, 46E22 |
| url | https://arxiv.org/abs/2601.18080 |