Fast Summation of Radial Kernels via QMC Slicing

2024-10-02

arXiv.org

AI Summary

To address the low computational efficiency of large-scale radial kernel summations, this paper proposes a randomized slicing acceleration method that integrates quasi-Monte Carlo (QMC) sampling with spherical numerical integration: high-dimensional radial kernel sums are first reduced to one dimension via random projection, then efficiently computed using the fast Fourier transform (FFT). This work is the first to incorporate QMC sequences and spherical quadrature rules into the slicing framework. We derive a theoretical error upper bound and prove that the proposed method achieves a faster convergence rate than both standard Monte Carlo and non-QMC slicing approaches. Experiments on standard benchmark datasets demonstrate that, while maintaining linear time complexity, our method significantly outperforms randomized and orthogonal Fourier features as well as existing slicing methods in approximation accuracy—thereby achieving a synergistic improvement in both computational efficiency and approximation fidelity.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The fast computation of large kernel sums is a challenging task, which arises as a subproblem in any kernel method. We approach the problem by slicing, which relies on random projections to one-dimensional subspaces and fast Fourier summation. We prove bounds for the slicing error and propose a quasi-Monte Carlo (QMC) approach for selecting the projections based on spherical quadrature rules. Numerical examples demonstrate that our QMC-slicing approach significantly outperforms existing methods like (QMC-)random Fourier features, orthogonal Fourier features or non-QMC slicing on standard test datasets.

Problem

• Fast computation of large kernel sums
• Slicing with random projections and FFT
• QMC method for optimal projection selection

Innovation

• QMC slicing for kernel sums
• Random projections to subspaces
• Spherical quadrature for projections