Discretization on high-dimensional domains

TL;DR

This paper develops a method for discretizing Lipschitz functions on high-dimensional metric spaces, providing bounds that outperform standard Monte Carlo methods, especially on spheres.

Contribution

It introduces a new discretization approach with explicit error bounds for Lipschitz functions on high-dimensional domains, including spheres, improving upon traditional Monte Carlo estimates.

Findings

Achieves error bounds of order N^{-1/2 - 3/(2β)} log^{1/2} N

Constant C is independent of the dimension d on spheres

Outperforms standard Monte Carlo error rates

Abstract

Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta>1$ such that $\mu(B) \geq c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $\Phi:[0,\infty)\to R$ that piecewisely lie in a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $\rho$ and the measure $\mu$ that for each positive integer $N\geq 2$, and each $g\in L^\infty(X, d\mu)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N}\in X$ and real numbers $\lambda_1, \ldots, \lambda_{ N}$ such that for any $x\in X$, \begin{align*} & \left| \int_X \Phi (\rho (x, y)) g(y) \,d \mu (y) - \sum_{j = 1}^{ N} \lambda_j \Phi (\rho (x, y_j)) \right| \leq C N^{- \frac{1}{2} - \frac{3}{2\beta}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $S^d$ of $R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac12}\sqrt{\log N}$.