Transformations and Hardy-Krause variation

TL;DR

This paper establishes conditions on transformations for functions to have bounded Hardy-Krause variation and to be smooth enough for advanced sampling methods, with applications to simplex and sphere transformations.

Contribution

It provides new criteria for transformations ensuring bounded variation and smoothness, enabling improved sampling accuracy in high-dimensional integration.

Findings

Certain transformations do not satisfy the bounded variation or smoothness conditions.

Some transformations satisfy only the bounded variation condition.

New transformations for the simplex improve sampling efficiency for generalized polynomials.

Abstract

Using a multivariable Faa di Bruno formula we give conditions on transformations $\tau:[0,1]^m\to\mathcal{X}$ where $\mathcal{X}$ is a closed and bounded subset of $\mathbb{R}^d$ such that $f\circ\tau$ is of bounded variation in the sense of Hardy and Krause for all $f\in C^d(\mathcal{x})$. We give similar conditions for $f\circ\tau$ to be smooth enough for scrambled net sampling to attain $O(n^{-3/2+\epsilon})$ accuracy. Some popular symmetric transformations to the simplex and sphere are shown to satisfy neither condition. Some other transformations due to Fang and Wang (1993) satisfy the first but not the second condition. We provide transformations for the simplex that makes $f\circ\tau$ smooth enough to fully benefit from scrambled net sampling for all $f$ in a class of generalized polynomials. We also find sufficient conditions for the Rosenblatt-Hlawka-M\"uck transformation in $\mathbb{R}^2$ and for importance sampling to be of bounded variation in the sense of Hardy and Krause.