破解高斯核的基函数有界性限制，揭示其逼近极限
Gaussian kernel expansion with basis functions uniformly bounded in $\mathcal{L}_{\infty}$
- 构造二维空间中高斯核的基函数有界展开
- 证明权重可属任意p>1的ℓ_p空间，但无法达到p=1
- 为核方法的泛化与收敛提供理论依据，适合理论研究者
核展开在机器学习中备受关注，尤其与特征映射密切相关。基函数与权重的性质（对应于Mercer分解中的特征函数与特征值）能揭示再生核希尔伯特空间结构、逼近效果、收敛速率及泛化能力。近期研究假设基函数在ℒ∞中一致有界，本文在此背景下系统研究高斯核的所有可能展开。主要成果是在ℝ²上构造出权重属于ℓ_p（任意p>1）的高斯核展开。该结果最优，因同时证明p=1不可达，且所有常用径向基函数核均无法实现。此外，该类核在ℝ²上不存在关于任意有限测度的Mercer展开，使得所有特征函数位于ℒ∞闭球内。
原文摘要 · Abstract (English)
Kernel expansions are a topic of considerable interest in machine learning, also because of their relation to the so-called feature maps introduced in machine learning. Properties of the associated basis functions and weights (corresponding to eigenfunctions and eigenvalues in the Mercer setting) give insight into for example the structure of the associated reproducing kernel Hilbert space, the goodness of approximation schemes, the convergence rates and generalization properties of kernel machines. Recent work in the literature has derived some of these results by assuming uniformly bounded basis functions in $\mathcal{L}_\infty$. Motivated by this line of research, we investigate under this constraint all possible kernel expansions of the Gaussian kernel, one of the most widely used models in machine learning. Our main result is the construction on $\mathbb{R}^2$ of a Gaussian kernel expansion with weights in $\ell_p$ for any $p>1$. This result is optimal since we also prove that $p=1$ cannot be reached by the Gaussian kernel, nor by any of the other radial basis function kernels commonly used in the literature. A consequence for this kind of kernels is also the non-existence of Mercer expansions on $\mathbb{R}^2$, with respect to any finite measure, whose eigenfunctions all belong to a closed ball of $\mathcal{L}_\infty$.
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