Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA

arXiv:2607.21823v1 Announce Type: new 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.

Related

• Dynamic Regret for Online Regression in RKHS via Discounted VAW and Subspace Approximation
• Which Spaces can be Embedded in L_p-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy
• New Bounds for Kernel Sums via Fast Spherical Embeddings

Source: arXiv cs.LG | 2026-07-27