We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on $\mathbb{R}^{d}$, including the strongly log-concave distributions under affine transformations. A feature of our result is that the data points need not have identical distributions or zero mean, which is key in certain applications such as clustering. Our bound for the Lipschitz kernels is dimension-free and sharp up to constants. For comparison, we also derive the companion result for the Euclidean (inner product) kernel for a class of sub-Gaussian distributions. A notable difference between the two cases is that, in contrast to the Euclidean kernel, in the Lipschitz case, the concentration inequality does not depend on the mean of the underlying vectors. As an application of these inequalities, we derive a bound on the misclassification rate of a kernel spectral clustering (KSC) algorithm, under a perturbed nonparametric mixture model. We show an example where this bound establishes the high-dimensional consistency (as $d\to \infty $) of the KSC, when applied with a Gaussian kernel, to a noisy model of nested nonlinear manifolds.

中文翻译：

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核矩阵的浓缩及其在核谱聚类中的应用

我们研究了随机核矩阵的均值附近的浓度。我们假设数据点是来自\ mathbb {R} ^ {d} $上的一类多元分布的独立绘图，包括仿射变换下的强对数凹面分布，则得出Lipschitz核的非渐近指数浓度不等式。结果的一个特点是数据点不必具有相同的分布或零均值，这在某些应用程序（例如聚类）中很关键。我们对Lipschitz内核的限制是无量纲的，并且可以精确到常量。为了进行比较，我们还针对一类次高斯分布推导了欧几里得（内积）内核的伴随结果。两种情况之间的显着差异是，与欧几里得核相比，在Lipschitz情况下，浓度不等式不取决于基础向量的均值。作为这些不等式的一种应用，我们在扰动的非参数混合模型下得出了核谱聚类（KSC）算法误分类率的界线。我们展示了一个示例，其中此边界在将高斯核应用于嵌套的非线性流形的噪声模型时，建立了KSC的高维一致性（如$ d \ to \ infty $）。