Approximate quadrature measures on data--defined spaces

TL;DR

This paper investigates conditions for approximate integration on complex spaces, demonstrating that quadrature formulas exact for diffusion polynomials satisfy specific error bounds, and explores kernel-based methods for constructing such formulas.

Contribution

It introduces a framework for approximate quadrature measures on measure spaces, generalizing recent results and connecting exactness for diffusion polynomials with error estimates.

Findings

Quadrature formulas exact for diffusion polynomials satisfy certain error bounds.

Kernel-based optimization can produce approximate quadrature formulas without exactness.

Results extend and unify recent advances in approximate integration on complex spaces.

Abstract

An important question in the theory of approximate integration is to study the conditions on the nodes $x_{k,n}$ and weights $w_{k,n}$ that allow an estimate of the form $$ \sup_{f\in \mathcal{B}_\gamma}|\sum_k w_{k,n}f(x_{k,n})-\int_\mathbb{X} fd\mu^*| \le cn^{-\gamma}, \qquad n=1,2,\cdots, $$ where $\mathbb{X}$ is often a manifold with its volume measure $\mu^*$, and $\mathcal{B}_\gamma$ is the unit ball of a suitably defined smoothness class, parametrized by $\gamma$. In this paper, we study this question in the context of a quasi-metric, locally compact, measure space $\mathbb{X}$ with a probability measure $\mu^*$. We show that quadrature formulas exact for integrating the so called diffusion polynomials of degree $<n$ satisfy such estimates. Without requiring exactness, such formulas can be obtained as a solutions of some kernel-based optimization problem. We discuss the connection with the question of optimal covering radius. Our results generalize in some sense many recent results in this direction.