This article proposes a first analysis of kernel spectral clustering methods
in the regime where the dimension $p$ of the data vectors to be clustered and
their number $n$ grow large at the same rate. We demonstrate, under a $k$-class
Gaussian mixture model, that the normalized Laplacian matrix associated with
the kernel matrix asymptotically behaves similar to a so-called spiked random
matrix. Some of the isolated eigenvalue-eigenvector pairs in this model are
shown to carry the clustering information upon a separability condition
classical in spiked matrix models. We evaluate precisely the position of these
eigenvalues and the content of the eigenvectors, which unveil important
(sometimes quite disruptive) aspects of kernel spectral clustering both from a
theoretical and practical standpoints. Our results are then compared to the
actual clustering performance of images from the MNIST database, thereby
revealing an important match between theory and practice.

本文通过对核谱聚类方法进行首次分析，发现在维度和数量同时增长的情况下，核矩阵的归一化拉普拉斯矩阵与所谓的尖峰随机矩阵呈类似的渐近行为。通过一种如尖峰矩阵模型的可分离条件，证明该模型中的一些孤立特征值 - 特征向量对携带聚类信息。我们精确评估了这些特征值的位置和特征向量内容，在理论和实践角度揭示了核谱聚类中非常重要（有时相当破坏性）的方面。最后将结果与 MNIST 数据库中图像实际聚类的性能进行比较，证明了理论和实践之间的重要匹配。