Spherical Cap $L_2$ Discrepancy -- Blessing of Dimensionality and a Balanced Large-Cap Variant
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| Format: | Preprint |
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2026
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| _version_ | 1866913057675935744 |
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| author | Brauchart, Johann S. Dick, Josef Pillichshammer, Friedrich |
| author_facet | Brauchart, Johann S. Dick, Josef Pillichshammer, Friedrich |
| contents | 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 $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21340 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spherical Cap $L_2$ Discrepancy -- Blessing of Dimensionality and a Balanced Large-Cap Variant Brauchart, Johann S. Dick, Josef Pillichshammer, Friedrich Numerical Analysis Number Theory 11K38, 65D30, 65C05, 65Y20 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 $d$. |
| title | Spherical Cap $L_2$ Discrepancy -- Blessing of Dimensionality and a Balanced Large-Cap Variant |
| topic | Numerical Analysis Number Theory 11K38, 65D30, 65C05, 65Y20 |
| url | https://arxiv.org/abs/2604.21340 |