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

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 $K(\mathbf{x},\mathbf{y})=1-\frac{1}{\sqrt{2}}\Vert \mathbf{x}-\mathbf{y}\Vert$. Stolarsky's invariance principle then implies that the worst-case integration error in this space grows polynomially with $d$.