A Multivariate Fast Discrete Walsh Transform with an Application to Function Interpolation

TL;DR

This paper introduces a fast multivariate discrete Walsh transform algorithm for high-dimensional function approximation and interpolation, enabling efficient computation on digital nets with applications to effective dimension estimation.

Contribution

It presents a novel $O(N \,\log N)$ algorithm for the discrete Walsh transform on digital nets, extending fast Fourier transform techniques to Walsh functions for high-dimensional data.

Findings

Efficient $O(N \log N)$ computation of Walsh coefficients

Construction of spline interpolants for high-dimensional functions

Application to estimating effective dimension in multivariate integration

Abstract

For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $O(N \log N)$ operations, where $N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented.