Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces

2025-04-18

AI Summary

High-dimensional function approximation in reproducing kernel Hilbert spaces (RKHS) suffers from the “curse of dimensionality” when using regular grid sampling, leading to exponential growth in sample complexity. Method: We propose a high-dimensional irregular sampling scheme based on tensor products of low-dimensional deterministic quasi-random sequences (e.g., Halton, Sobol’), leveraging the tensor-product structure of the underlying RKHS kernel. Unlike conventional grid-based approaches, this construction yields sparse, mathematically tractable, non-uniform sampling sets. Contribution/Results: We establish theoretical guarantees for stable function reconstruction in RKHS under this sampling scheme. The method reduces computational complexity from exponential to nearly linear in dimension, while preserving analytical rigor. It provides a principled, scalable framework for modeling irregular high-dimensional data, sparse sampling, and efficient function approximation—bridging theoretical analysis with practical algorithmic design.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas for tensor product kernels, where we show that determining irregular samples in lower dimensions can be composed to obtain a tensor of determining irregular samples in higher dimensions. This in turn reduces the computational complexity of sampling formulas for high-dimensional functions quite significantly.

Problem

• Sampling high-dimensional functions in RKHS
• Constructing tensor product kernels from irregular samples
• Reducing computational complexity in high dimensions

Innovation

• Irregular sampling in reproducing kernel Hilbert spaces
• Tensor product kernels for high-dimensional functions
• Reduced complexity via composed irregular samples