Spectral Properties of Radial Kernels and Clustering in High Dimensions
This work addresses clustering challenges in high-dimensional data analysis, offering a theoretical foundation for kernel-based methods, but it appears incremental as it builds on existing spectral techniques with specific kernel adaptations.
The paper tackles the problem of clustering high-dimensional mixtures with possibly common means but different covariance matrices by analyzing the spectral properties of radial kernels, and it shows that a specific kernel enables a simple spectral algorithm with a required minimum angular separation between covariance matrices that tends to zero as dimensionality increases.
In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in $\mathbb{R}^n$. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but different covariance matrices. We show that the minimum angular separation between the covariance matrices that is required for the algorithm to succeed tends to $0$ as $n$ goes to infinity.