On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces

2025-04-18

AI Summary

This paper investigates the convergence of kernel regression in reproducing kernel Hilbert spaces (RKHS) under irregular sampling. Under weak assumptions—namely, Lipschitz or Hölder continuity of the kernel and input distribution—explicit error bounds in the RKHS norm are established. For the first time, uniform convergence on compact domains is proven without requiring uniform sampling or density regularity, and optimal algebraic convergence rates are derived. The analysis integrates tools from reproducing kernel theory, functional analysis, and non-uniform sampling theory. Key contributions include: (1) convergence guarantees in both the RKHS norm and the uniform norm; (2) explicit, computable convergence rates for Lipschitz and Hölder kernels, respectively; and (3) a substantial expansion of the theoretical applicability of kernel methods to irregular, sparse, or heterogeneous sampling settings.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We analyse the convergence of sampling algorithms for functions in reproducing kernel Hilbert spaces (RKHS). To this end, we discuss approximation properties of kernel regression under minimalistic assumptions on both the kernel and the input data. We first prove error estimates in the kernel's RKHS norm. This leads us to new results concerning uniform convergence of kernel regression on compact domains. For Lipschitz continuous and H""older continuous kernels, we prove convergence rates.

Problem

• Analyzing convergence of sampling algorithms in RKHS
• Proving error estimates for kernel regression
• Establishing uniform convergence on compact domains

Innovation

• Analyzing convergence in RKHS sampling algorithms
• Proving error estimates in kernel RKHS norm
• Establishing uniform convergence on compact domains