论文标题
$ \ varepsilon $ - 内核类别的非反应光谱界
Non-asymptotic spectral bounds on the $\varepsilon$-entropy of kernel classes
论文作者
论文摘要
令$ k：\boldsymbolΩ\ times \boldsymbolΩ$为$ {\ Mathbb r}^n $的紧凑子集定义的连续Mercer内核，以及$ \ Mathcal {h} _k $是与$ k $相关的重现Kernel Hilbert Space（RKHS）。给定$ \boldsymbolΩ$上的有限度量$ν$，我们调查了$ \ MATHCAL {H} _K $的$ \ varepsilon $ -Entropy上的上限和下限。该主题是基于内核方法的现代统计理论的重要方向。 我们证明了[1，+\ infty] $的$ p \的上限和下限。对于$ p \在[1,2] $中，上限仅取决于相应的积分操作员$ ϕ \ to \ int _ {\boldsymbolΩ} k（\ cdot，{\ mathbf y}）ϕ（{\ mathbf y}）d C（\ mathb）在约束中，对于$ p> 2 $，界限还取决于截断的Mercer系列的收敛速率与$ L_P（ν）$ - NORM中的内核$ K $。 我们讨论了界限的许多后果，并表明它们比一般内核的先前界限要高得多。此外，对于特定情况，例如区域内核和盒子上的高斯内核，我们的边界渐近地紧密，如$ \ varepsilon \ to +0 $。
Let $K: \boldsymbolΩ\times \boldsymbolΩ$ be a continuous Mercer kernel defined on a compact subset of ${\mathbb R}^n$ and $\mathcal{H}_K$ be the reproducing kernel Hilbert space (RKHS) associated with $K$. Given a finite measure $ν$ on $\boldsymbolΩ$, we investigate upper and lower bounds on the $\varepsilon$-entropy of the unit ball of $\mathcal{H}_K$ in the space $L_p(ν)$. This topic is an important direction in the modern statistical theory of kernel-based methods. We prove sharp upper and lower bounds for $p\in [1,+\infty]$. For $p\in [1,2]$, the upper bounds are determined solely by the eigenvalue behaviour of the corresponding integral operator $ϕ\to \int_{\boldsymbolΩ} K(\cdot,{\mathbf y})ϕ({\mathbf y})dν({\mathbf y})$. In constrast, for $p>2$, the bounds additionally depend on the convergence rate of the truncated Mercer series to the kernel $K$ in the $L_p(ν)$-norm. We discuss a number of consequences of our bounds and show that they are substantially tighter than previous bounds for general kernels. Furthermore, for specific cases, such as zonal kernels and the Gaussian kernel on a box, our bounds are asymptotically tight as $\varepsilon\to +0$.
