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
This work addresses theoretical analysis challenges in complex systems and machine learning, though it appears incremental as it builds on existing operator theory and residual analysis concepts.
The paper tackles the problem of analyzing complex systems with component subdivisions by developing an operator theoretic framework for residual analysis, proving new admissibility results and a priori bounds on energy residuals. It applies this framework to machine learning algorithms like greedy Kernel PCA, providing explicit convergence results, residual energy decomposition, and stability criteria under noise.
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.