In this paper we study the concentration properties for the eigenvalues of
kernel matrices, which are central objects in a wide range of kernel methods
and, more recently, in network analysis. We present a set of concentration
inequalities tailored for each individual eigenvalue of the kernel matrix with
respect to its known asymptotic limit. The inequalities presented here are of
relative type, meaning that they scale with the eigenvalue in consideration,
which results in convergence rates that vary across the spectrum. The rates we
obtain here are faster than the typical $\O(\frac{1}{\sqrt n})$ and are often
exponential, depending on regularity assumptions of Sobolev type. One key
feature of our results is that they apply to non positive kernels, which is
fundamental in the context of network analysis. We show how our results are
well suited for the study of dot product kernels, which are related to random
geometric graphs on the sphere, via the graphon formalism. We illustrate our
results by applying them to a variety of dot product kernels on the sphere and
to the one dimensional Gaussian kernel.

本文研究了核矩阵特征值的集中性质，证明了这些集中不等式适用于各种核矩阵的特征值，特别是非正定核矩阵在网络分析中的应用。

核矩阵谱的相对浓度界限

Relative concentration bounds for the spectrum of kernel matrices

We study concentration properties of random vectors of the form $AX$, where
$X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. We
show that the distribution of $AX$ is well spread in space whenever the
distributions of $X_i$ are well spread on the line. Specifically, assume that
the probability that $X_i$ falls in any given interval of length $T$ is at most
$p$. Then the probability that $AX$ falls in any given ball of radius $T
\|A\|_{HS}$ is at most $(Cp)^{0.9 r(A)}$, where $r(A)$ denotes the stable rank
of $A$ and $C$ is an absolute constant.

研究随机向量的集中性质，其中 X =（X1，...，Xn）具有独立坐标，而 A 是给定的矩阵。我们证明，只要 X 的分布在线上很好地分布，AX 的分布就会在空间中得到很好的展开。