论文标题
对具有较小平滑度的功能类别的积分规范进行样本离散误差
Sampling discretization error of integral norms for function classes with small smoothness
论文作者
论文摘要
我们考虑了函数的无限尺寸类别，而不是相对误差设置，而不是相对误差设置，该设置用于积分规范离散化的先前论文中，我们考虑了绝对误差设置。我们展示了如何将研究理论和数值整合的两个领域的已知结果用于对不同函数类别的平方规范的离散化。我们证明了一个一般结果，该结果表明，在某种意义上说，统一规范中函数类的熵数序列占主导地位，是该类别平方规范的离散化误差序列。然后，我们使用此结果来建立新的误差范围，以在具有混合平滑度的多元函数类别上取消平方规范的离散化。
We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research -- supervised learning theory and numerical integration -- can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of multivariate functions with mixed smoothness.
