提出新算子理论框架，分析多通道信号残差与能量分解。
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
- 基于算子缺陷恒等式构建残差分析新框架。
- 给出能量残差的先验界与收敛性证明。
- 适用于机器学习中的核方法与噪声稳定性分析。
我们提出一种新的算子理论框架，用于分析具有内在分量结构的复杂系统，形式上表现为一般残差及特定的级联能量残差。证明了新的可适配性/有效性结果，以及能量残差的先验界。应用包括无穷维Kaczmarz理论中λₙ-松弛变体及其λₙ-有效性分析；并应用于广义机器学习算法，如贪婪核主成分分析（KPCA），证明了显式的收敛结果、残差能量分解，以及在噪声下的稳定性判据。
原文摘要 · Abstract (English)
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.
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