Adaptive Kernel Selection for Kernelized Diffusion Maps

• URL: http://arxiv.org/abs/2604.18402v1
• Date: Mon, 20 Apr 2026 15:22:46 GMT
• Title: Adaptive Kernel Selection for Kernelized Diffusion Maps
• Abstract summary: We introduce two complementary approaches to adaptive kernel selection for Diffusion Maps.<n>First, we develop a variational outer loop that learns continuous kernel parameters, including bandwidths and mixture weights.<n>Second, we propose an unsupervised cross-validation pipeline that selects kernel families and bandwidths using an eigenvalue-sum.
• Score: 1.7499351967216341
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Selecting an appropriate kernel is a central challenge in kernel-based spectral methods. In \emph{Kernelized Diffusion Maps} (KDM), the kernel determines the accuracy of the RKHS estimator of a diffusion-type operator and hence the quality and stability of the recovered eigenfunctions. We introduce two complementary approaches to adaptive kernel selection for KDM. First, we develop a variational outer loop that learns continuous kernel parameters, including bandwidths and mixture weights, by differentiating through the Cholesky-reduced KDM eigenproblem with an objective combining eigenvalue maximization, subspace orthonormality, and RKHS regularization. Second, we propose an unsupervised cross-validation pipeline that selects kernel families and bandwidths using an eigenvalue-sum criterion together with random Fourier features for scalability. Both methods share a common theoretical foundation: we prove Lipschitz dependence of KDM operators on kernel weights, continuity of spectral projectors under a gap condition, a residual-control theorem certifying proximity to the target eigenspace, and exponential consistency of the cross-validation selector over a finite kernel dictionary.

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