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Gaussian RBF Centered Kernel Alignment (CKA) in the Large-Bandwidth Limit.

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    Summary
    This summary is machine-generated.

    Centered Kernel Alignment (CKA) using Gaussian RBF kernels converges to linear CKA with increasing bandwidth. Representation geometry, specifically eccentricity (ρ), dictates the bandwidth range where nonlinearities are preserved. Select bandwidths below ρ for nonlinear modeling.

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    Area of Science:

    • Machine Learning
    • Kernel Methods
    • Representation Learning

    Background:

    • Centered Kernel Alignment (CKA) is a key metric for comparing feature representations and kernels.
    • Understanding the behavior of CKA with different kernel types is crucial for its effective application.

    Purpose of the Study:

    • To analyze the convergence of Gaussian RBF kernel CKA to linear CKA.
    • To investigate the influence of feature representation geometry on this convergence.

    Main Methods:

    • Theoretical analysis of CKA with Gaussian RBF kernels in the large-bandwidth limit.
    • Utilizing mean-centering and Hilbert-Schmidt Independence Criterion (HSIC) identities.
    • Introducing and analyzing 'representation eccentricity' (ρ).

    Main Results:

    • Proved that Gaussian RBF kernel CKA converges to linear CKA as bandwidth increases.
    • Demonstrated that the convergence onset depends on representation geometry (ρ).
    • Showed that Gaussian CKA differs from linear CKA within a bandwidth range constrained by ρ.

    Conclusions:

    • Gaussian CKA behavior is sensitive to feature representation geometry.
    • Bandwidths smaller than representation eccentricity (ρ) are recommended for capturing nonlinearities.
    • This provides guidance for selecting appropriate bandwidths in kernel-based representation similarity analysis.