Median of Means Sampling for the Keister Function

Abstract

This study investigates the performance of median-of-means sampling compared to traditional mean-of-means sampling for computing the Keister function integral using Randomized Quasi-Monte Carlo (RQMC) methods. The research tests both lattice points and digital nets as point distributions across dimensions 2, 3, 5, and 8, with sample sizes ranging from 2^8 to 2^19 points. Results demonstrate that median-of-means sampling consistently outperforms mean-of-means for sample sizes larger than 10^3 points, while mean-of-means shows better accuracy with smaller sample sizes, particularly for digital nets. The study also confirms previous theoretical predictions about median-of-means' superior performance with larger sample sizes and reflects the known challenges of maintaining accuracy in higher-dimensional integration. These findings support recent research suggesting median-of-means as a promising alternative to traditional sampling methods in numerical integration, though limitations in sample size and dimensionality warrant further investigation with different test functions and larger parameter spaces.

Figures (6)

• Figure 1: The effects of outliers the mean and median of a nearly normal distribution
• Figure 2: Left: a 2-dimensional lattice, right: a 2-dimensional digital net. Both point sets contain 64 points and are generated by QMCPy.
• Figure 3: Comparison of digital net generators in dimensions 2,3,5,8. Results can be conditionally reproduced.
• Figure 4: Difference graph of digital net generators in dimensions 2,3,5,8. Results can be conditionally reproduced.
• Figure 5: Comparison of lattice generators in dimensions 2,3,5,8. Results can be conditionally reproduced.