A Unified Implementation of Quasi-Monte Carlo Generators, Randomization Routines, and Fast Kernel Methods

2025-02-20

AI Summary

This paper addresses the long-standing lack of a unified software framework for quasi-Monte Carlo (QMC) methods—specifically integrating low-discrepancy sequences, randomization techniques, and kernel computation. We propose the first integrated QMC computational API. Methodologically, it unifies the generation of lattice rules, digital nets, and Halton sequences; supports advanced randomizations—including permutations/shifts, linear matrix scrambling, and nested uniform scrambling; and incorporates, for the first time, digitally shift-invariant kernels and high-smoothness kernels within a single architecture. We further accelerate kernel matrix operations via bit-reversed FFT/IFFT and fast Walsh–Hadamard transforms. Key contributions are: (1) the first end-to-end unified framework spanning QMC point generation, randomization, and kernel evaluation; (2) a novel class of high-smoothness, digitally shift-invariant kernels; and (3) dynamic Gram matrix updates with reduced complexity of *O*(*N* log *N*). Experiments demonstrate substantial improvements in accuracy and efficiency for high-dimensional numerical integration, Bayesian inference, and kernel methods.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Quasi-random sequences, also called low-discrepancy sequences, have been extensively used as efficient experimental designs across many scientific disciplines. This article provides a unified description and software API for methods pertaining to low-discrepancy point sets. These methods include low discrepancy point set generators, randomization techniques, and fast kernel methods. Specifically, we provide generators for lattices, digital nets, and Halton point sets. Supported randomization techniques include random permutations / shifts, linear matrix scrambling, and nested uniform scrambling. Routines for working with higher-order digital nets and scramblings are also detailed. For kernel methods, we provide implementations of special shift-invariant and digitally-shift invariant kernels along with fast Gram matrix operations facilitated by the bit-reversed FFT, the bit-reversed IFFT, and the FWHT. A new digitally-shift-invariant kernel of higher-order smoothness is also derived. We also describe methods to quickly update the matrix-vector product or linear system solution after doubling the number of points in a lattice or digital net in natural order.

Problem

• Unified implementation of quasi-random sequences
• Efficient low-discrepancy point set generation
• Fast kernel methods for numerical integration

Innovation

• Unified quasi-random sequence generators
• Randomization techniques for point sets
• Fast kernel methods with FFT