接近比专注更重要？光滑径向核推理的一种近似观点
Approximation beats concentration? An approximation view on inference with smooth radial kernels
摘要：
正定核及其相关的再生核希尔伯特空间为从数据中学习提供了一个数学上令人信服且实际上具有竞争力的框架。在本文中，我们采取近似理论的观点来探讨光滑核的各个方面与它们的推理性质。我们分析了核算子和矩阵的特征值衰减，特征函数/特征向量的性质和“傅立叶”系数的功能限制在一个离散的数据点的核空间。我们还研究了核的拟合能力，给出了再生核希尔伯特空间中球的脂肪粉碎维数的明确界限。有趣的是，使核函数在其“原生”核空间中非常有效的逼近器的相同属性，也限制了它们表示任意函数的能力。我们讨论了各种影响，包括那些梯度下降型方法。值得注意的是，我们的大多数边界都是测度独立的。此外，至少在中等维度上，特征值的界限比通常的矩阵集中结果所能得到的界限要严格得多。例如，我们看到核矩阵的特征值显示出近似指数衰减，常数仅取决于核和域。我们称之为“近似节拍集中”现象，因为即使数据是从概率分布中采样的，它们的一些方面在近似理论中也能得到更好的理解。
英文摘要：
Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential properties. We analyze eigenvalue decay of kernels operators and matrices, properties of eigenfunctions/eigenvectors and "Fourier" coefficients of functions in the kernel space restricted to a discrete set of data points. We also investigate the fitting capacity of kernels, giving explicit bounds on the fat shattering dimension of the balls in Reproducing Kernel Hilbert spaces. Interestingly, the same properties that make kernels very effective approximators for functions in their "native" kernel space, also limit their capacity to represent arbitrary functions. We discuss various implications, including those for gradient descent type methods. It is important to note that most of our bounds are measure independent. Moreover, at least in moderate dimension, the bounds for eigenvalues are much tighter than the bounds which can be obtained from the usual matrix concentration results. For example, we see that the eigenvalues of kernel matrices show nearly exponential decay with constants depending only on the kernel and the domain. We call this "approximation beats concentration" phenomenon as even when the data are sampled from a probability distribution, some of their aspects are better understood in terms of approximation theory.
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