统一多种核函数，构建可灵活切换支持范围的通用核类。
Unified Native Spaces in Kernel Methods
- 提出一个统一框架，将众多经典核函数作为特例纳入其中。
- 新核对应的RKHS与Sobolev空间等价，可推导已有核的空间性质。
- 支持从紧支到全局支撑切换，且可在非平凡区间取负值，适合复杂建模。
正定核的参数化模型种类繁多，在统计学、机器学习、数值分析和逼近论等领域广泛应用。通常，核参数对应于相关过程的某些特征，如光滑性（基于Sobolev空间、均方可微性及分形维数）、紧支或全局支集、负相关性（空洞效应）等，这些特征对多个理论与应用领域具有重要意义。本文将大量已知核函数统一为单一参数化类，该类通过精确参数化或参数渐近实现对它们的涵盖。我们进一步刻画了与新核对应的RKHS在范数上等价的Sobolev空间，并由此推导出现有各类核所关联的Sobolev空间。通过实例展示了新类核的主要性质，说明其可实现从紧支到全局支集的切换，并给出了在非平凡区间取负值的特殊情形。因此，所提出的核类是包含诸多特例（包括著名的Matérn与Wendland核及其带空洞效应的变体）的丰富希尔伯特空间的再生核。
原文摘要 · Abstract (English)
There exists a plethora of parametric models for positive definite kernels, and their use is ubiquitous in disciplines as diverse as statistics, machine learning, numerical analysis, and approximation theory. Usually, the kernel parameters index certain features of an associated process. Amongst those features, smoothness (in the sense of Sobolev spaces, mean square differentiability, and fractal dimensions), compact or global supports, and negative dependencies (hole effects) are of interest to several theoretical and applied disciplines. This paper unifies a wealth of well-known kernels into a single parametric class that encompasses them as special cases, attained either by exact parameterization or through parametric asymptotics. We furthermore characterize the Sobolev space that is norm equivalent to the RKHS associated with the new kernel. As a by-product, we infer the Sobolev spaces that are associated with existing classes of kernels. We illustrate the main properties of the new class, show how this class can switch from compact to global supports, and provide special cases for which the kernel attains negative values over nontrivial intervals. Hence, the proposed class of kernel is the reproducing kernel of a very rich Hilbert space that contains many special cases, including the celebrated Matérn and Wendland kernels, as well as their aliases with hole effects.
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