This paper is a follow-up to the recent paper "A note on isotropic discrepancy and spectral test of lattice point sets" [J. Complexity, 58:101441, 2020]. We show that the isotropic discrepancy of a lattice point set is at most $d \, 2^{2(d+1)}$ times its spectral test, thereby correcting the dependence on the dimension $d$ and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube.
Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test -- and with it the isotropic discrepancy -- is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.
QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces
Institut für Mathematik, Universität Osnabrück（奥斯纳布吕克大学）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学）；Stuttgart Technology Center, Sony Semiconductor Solutions Europe（索尼半导体解决方案欧洲斯图加特技术中心）；Mathematisches Institut, Friedrich-Schiller-Universität Jena（耶拿大学）
Deep ReLU neural networks in high-dimensional approximation
我们研究了深度ReLU（Rectified Linear Unit）神经网络在$d$维单位立方体上定义的混合光滑的Hölder-Zygmund空间中逼近函数的计算复杂性，当维度$d$可能非常大时。逼近误差以各向同性索伯列夫空间的范数度量。对于来自混合光滑的Hölder-Zygmund空间的每个函数$f$，我们显式地构建了一个深度ReLU神经网络，其输出以规定的精度$\varepsilon$近似$f$，并证明了这种近似的计算复杂性的严格的维度依赖的上限和下界，其特征在于这种深度ReLU神经网络的大小和深度，显式地在$d$和$\varepsilon$中。这些结果的证明特别依赖于基于Faber级数的稀疏网格采样恢复的近似。
