The performance of spectral clustering relies on the fluctuations of the
entries of the eigenvectors of a similarity matrix, which has been left
uncharacterized until now. In this letter, it is shown that the signal $+$
noise structure of a general spike random matrix model is transferred to the
eigenvectors of the corresponding Gram kernel matrix and the fluctuations of
their entries are Gaussian in the large-dimensional regime. This CLT-like
result was the last missing piece to precisely predict the classification
performance of spectral clustering. The proposed proof is very general and
relies solely on the rotational invariance of the noise. Numerical experiments
on synthetic and real data illustrate the universality of this phenomenon.

通过对相似性矩阵的特征向量的波动性进行建模，证明了在大维空间中其元素的波动服从高斯分布，从而精确预测了谱聚类的分类性能。通过对合成数据和真实数据的数值实验，证明了这一现象的普适性。

谱聚类中特征向量的渐近高斯波动

Asymptotic Gaussian Fluctuations of Eigenvectors in Spectral Clustering

We study products of random matrices in the regime where the number of terms
and the size of the matrices simultaneously tend to infinity. Our main theorem
is that the logarithm of the $\ell_2$ norm of such a product applied to any
fixed vector is asymptotically Gaussian. The fluctuations we find can be
thought of as a finite temperature correction to the limit in which first the
size and then the number of matrices tend to infinity. Depending on the scaling
limit considered, the mean and variance of the limiting Gaussian depend only on
either the first two or the first four moments of the measure from which matrix
entries are drawn. We also obtain explicit error bounds on the moments of the
norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our
result to obtain precise information about the stability of gradients in
randomly initialized deep neural networks with ReLU activations. This provides
a quantitative measure of the extent to which the exploding and vanishing
gradient problem occurs in a fully connected neural network with ReLU
activations and a given architecture.

研究随机矩阵的乘积，证明其对于任意固定向量的 2 范数的对数渐近于高斯分布，并将其应用于测量深度神经网络的激活函数 ReLU 下的梯度稳定性问题。