论文标题
朝向矢量值函数的经验过程理论：平滑函数类的度量熵
Towards Empirical Process Theory for Vector-Valued Functions: Metric Entropy of Smooth Function Classes
论文作者
论文摘要
本文为开发经验过程理论提供了一些第一步，以在矢量空间中占据值的功能。我们的主要结果通过利用矢量值函数的差分计算和度量空间的分形维理论来利用理论来利用理论，从而在希尔伯特空间中获得值的平滑函数类别的熵。我们演示了这些熵界如何用于显示函数类别的统一定律和渐近等级的统一定律，并将其应用于统计学习理论，其中输出空间是希尔伯特空间。最后，我们讨论了Rademacher复杂性向矢量值函数类别的扩展。
This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by leveraging theory from differential calculus of vector-valued functions and fractal dimension theory of metric spaces. We demonstrate how these entropy bounds can be used to show the uniform law of large numbers and asymptotic equicontinuity of the function classes, and also apply it to statistical learning theory in which the output space is a Hilbert space. We conclude with a discussion on the extension of Rademacher complexities to vector-valued function classes.
