论文标题
$ l_2 $ - 纳米采样离散和从RKHS恢复有限跟踪的功能
$L_2$-norm sampling discretization and recovery of functions from RKHS with finite trace
论文作者
论文摘要
在本文中，我们研究了基于随机函数样本的$ d \ subset \ r^d $在RKHS上的复杂值函数的$ L_2 $ -Norm采样和采样恢复。我们仅假设内核的有限跟踪（Hilbert-Schmidt嵌入$ L_2 $），并为相应的最坏情况错误提供了几个具有精确常数的具体估计。通常，我们的分析不需要任何其他假设，还包括非碳核和不可分割的RKHS的情况。失败概率受到控制，并以$ n $（样本数量）为单位衰减。在轻度的可分离性假设下，我们观察到与奇异值衰减相关的收敛速率提高。我们的主要工具是针对鲁德尔森，门德尔森，帕乔尔，奥利维拉和劳胡特的无限复合随机矩阵的光谱范围浓度不等式。
In this paper we study $L_2$-norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $D \subset \R^d$ based on random function samples. We only assume the finite trace of the kernel (Hilbert-Schmidt embedding into $L_2$) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in $n$, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.
