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Gaussian RBF Kernels Asymptotically Converge to Linear Kernels.

Sergio A. Alvarez· July 27, 2026 View original

Summary

This paper demonstrates that Gaussian Radial Basis Function (RBF) reproducing kernel Hilbert spaces (RKHS) become asymptotically isometric to Euclidean space in the large bandwidth limit. This implies that kernel-based methods using Gaussian RBFs will behave like linear kernels with large bandwidths, affecting techniques like Kernel PCA.

This research explores the asymptotic behavior of Gaussian Radial Basis Function (RBF) kernels, a common component in many machine learning algorithms. The study reveals that, as the bandwidth of the Gaussian RBF increases, its associated reproducing kernel Hilbert space (RKHS) becomes increasingly similar to Euclidean space, effectively behaving like a linear kernel. This finding has significant implications for understanding how kernel-based constructions, particularly those relying on metric properties of the RKHS, will perform under large bandwidths. The paper specifically examines Kernel PCA, showing that its eigenvalues, eigenprojections, and principal components converge to those of classical (linear) PCA as the bandwidth approaches infinity. The difference between the RKHS feature embeddings and the orthogonal PCA eigenframes of Gaussian RBF and linear kernels asymptotically reduces to a geometric similarity transformation, with a residual term dependent on the data's geometric eccentricity. This eccentricity, measured by the ratio of maximum to median pairwise data distances, serves as a reliable predictor for dataset-specific convergence in the principal directions.

Why it matters

Data scientists and ML engineers working with kernel methods need to understand how kernel parameters, especially bandwidth, influence model behavior and computational complexity, potentially simplifying model selection and interpretation.

How to implement this in your domain

  1. 1Optimize Gaussian RBF kernel bandwidths, considering the trade-off between non-linearity and linear approximation.
  2. 2Evaluate the geometric eccentricity of datasets to predict the convergence behavior of Kernel PCA.
  3. 3Simplify model architectures by using linear kernels when large Gaussian RBF bandwidths are found to be optimal.
  4. 4Develop diagnostic tools to visualize and analyze the RKHS properties and their convergence to Euclidean space.

Who benefits

Data ScienceMachine Learning EngineeringResearch & DevelopmentAI Consulting

Key takeaways

  • Gaussian RBF kernels behave increasingly like linear kernels as their bandwidth increases.
  • This convergence impacts kernel-based methods, including Kernel PCA, which approaches classical PCA.
  • Data eccentricity can predict how quickly Gaussian RBF kernel methods converge to their linear counterparts.
  • Understanding this asymptotic behavior can guide hyperparameter tuning and model simplification.

Original post by Sergio A. Alvarez

"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 reli…"

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Originally posted by Sergio A. Alvarez on X · view source

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