Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA
TL;DR
It is shown that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit, suggesting that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths.
Abstract
We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $\sigma \rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\frac{\rho}{\sigma} \right )^2$, where $\rho$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $\rho$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.