In this paper, we study the convergence of the spectral embeddings obtained from the leading eigenvectors of certain similarity matrices to their population counterparts. We opt to study this convergence in a uniform (instead of average) sense and highlight the benefits of this choice. Using the Newton-Kantorovich Theorem and other tools from functional analysis, we first establish a general perturbation result for orthonormal bases of invariant subspaces. We then apply this general result to normalized spectral clustering. By tapping into the rich literature of Sobolev spaces and exploiting some concentration results in Hilbert spaces, we are able to prove a finite sample error bound on the uniform consistency error of the spectral embeddings in normalized spectral clustering.

翻译：本文研究了由某些相似性矩阵的主特征向量所获得的谱嵌入向总体谱嵌入收敛的问题。我们选择从一致（而非平均）意义上研究这种收敛性，并强调了这一选择的优势。利用牛顿-康托罗维奇定理以及其他泛函分析工具，我们首先建立了一个关于不变子空间标准正交基的一般性扰动结果。随后，我们将这一一般性结果应用于归一化谱聚类。通过利用索博列夫空间中丰富的文献资料，并借助希尔伯特空间中的一些集中性结果，我们得以证明归一化谱聚类中谱嵌入的一致收敛误差的有限样本误差界。