Optimizing Kernel Discrepancies via Subset Selection

2025-11-04

AI Summary

This work addresses the problem of efficiently selecting an $m$-element subset from a large candidate set to minimize kernel discrepancy, thereby reducing worst-case error in quasi-Monte Carlo (QMC) methods for high-dimensional numerical integration and uncertainty quantification. The proposed method introduces a general, kernel-discrepancy-guided greedy subset selection algorithm applicable to both the uniform distribution on the unit hypercube and arbitrary distributions $F$ with known density functions. It constitutes the first extension of kernel Stein discrepancy to deterministic sampling under non-uniform distributions and establishes novel theoretical connections between $L_2$-star discrepancy and $L_infty$-discrepancy. Empirical evaluation demonstrates that the approach significantly reduces integration error in high dimensions and consistently outperforms state-of-the-art QMC and random sampling strategies across diverse target distributions.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Kernel discrepancies are a powerful tool for analyzing worst-case errors in quasi-Monte Carlo (QMC) methods. Building on recent advances in optimizing such discrepancy measures, we extend the subset selection problem to the setting of kernel discrepancies, selecting an m-element subset from a large population of size $n gg m$. We introduce a novel subset selection algorithm applicable to general kernel discrepancies to efficiently generate low-discrepancy samples from both the uniform distribution on the unit hypercube, the traditional setting of classical QMC, and from more general distributions $F$ with known density functions by employing the kernel Stein discrepancy. We also explore the relationship between the classical $L_2$ star discrepancy and its $L_infty$ counterpart.

Problem

• Extends subset selection to kernel discrepancies for quasi-Monte Carlo analysis
• Introduces algorithm to generate low-discrepancy samples from uniform and general distributions
• Explores relationship between classical L2 star discrepancy and L∞ counterpart

Innovation

• Subset selection algorithm optimizes kernel discrepancies
• Generates low-discrepancy samples from uniform distributions
• Extends to general distributions using kernel Stein discrepancy