Spherical Cap $L_2$ Discrepancy -- Blessing of Dimensionality and a Balanced Large-Cap Variant

TL;DR

This paper demonstrates a dimensionality benefit for spherical cap $L_2$ discrepancy in numerical integration, introduces a large-cap variant that does not become easier with higher dimensions, and connects it to Sobolev space integration errors.

Contribution

It proves a blessing of dimensionality for classical discrepancy, introduces a large-cap variant, and establishes a Stolarsky invariance principle linking discrepancy to Sobolev space integration.

Findings

Information complexity decreases with dimension for classical discrepancy.

Large-cap discrepancy does not become easier as dimension increases.

Worst-case integration error grows polynomially with dimension.

Abstract

We prove that the information complexity (i.e., the inverse) of the classical spherical cap $L_2$ discrepancy on the $d$-dimensional sphere $\mathbb{S}^d$ decreases with dimension $d$, indicating a ``blessing of dimensionality'' for the associated numerical integration problem. We then introduce a modified spherical cap $L_2$ discrepancy that emphasizes large caps (close to hemispheres). For this variant, the problem does not become easier with increasing $d$. We also establish a Stolarsky invariance principle which connects the modified spherical cap $L_2$ discrepancy to numerical integration in the Sobolev space $H^{(d+1)/2}(\mathbb{S}^d)$, represented by the reproducing kernel $K(\boldsymbol{x}, \boldsymbol{y}) = 1 - \tfrac{1}{\sqrt{2}} \|\boldsymbol{x} - \boldsymbol{y}\|$. Stolarsky's invariance principle then implies that the worst-case integration error in this space grows polynomially with $d$.