论文标题
高斯和相关分析核的希尔伯特空间中的近似
Approximation in Hilbert spaces of the Gaussian and related analytic kernels
论文作者
论文摘要
我们考虑基于功能评估的线性近似值，以重现某些分析加权功率系列内核的内核希尔伯特空间和间隔$ [-1,1] $的固定核。这两个类都包含流行的高斯内核$ k（x，y）= \ exp（ - \ tfrac {1} {2} {2} \ varepsilon^2（x-y）^2）$。对于加权功率系列内核，我们在最坏情况下的误差上得出了几乎匹配的上和下限。当应用于高斯内核时，我们的结果指出，最多可为子指数因素，最小误差衰减为$（\ varepsilon/n）^n（n！）^{ - 1/2} $。证明基于加权多项式插值和经典多项式系数估计。
We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = \exp(-\tfrac{1}{2}\varepsilon^2(x-y)^2)$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel, our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(\varepsilon/n)^n (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates.
