Adaptive Kernel Selection for Kernelized Diffusion Maps
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914492229615616 |
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| author | Aboussaad, Othmane Miraoui, Adam Hamzi, Boumediene Owhadi, Houman |
| author_facet | Aboussaad, Othmane Miraoui, Adam Hamzi, Boumediene Owhadi, Houman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18402 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adaptive Kernel Selection for Kernelized Diffusion Maps Aboussaad, Othmane Miraoui, Adam Hamzi, Boumediene Owhadi, Houman Machine Learning Dynamical Systems 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. |
| title | Adaptive Kernel Selection for Kernelized Diffusion Maps |
| topic | Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2604.18402 |