# Adaptive Kernel Selection for Kernelized Diffusion Maps

arXiv: https://arxiv.org/abs/2604.18402 · HTML: https://mlsanity.com/paper/2604.18402/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 42 references that are in the map.
- First posted on arXiv 20 Apr 2026.

## Builds on (most cited first, 19 of 19)

- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems](https://mlsanity.com/paper/2301.10321/) (2023)
- [Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem](https://mlsanity.com/paper/2012.00338/) (2020)
- [Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds](https://mlsanity.com/paper/1904.08622/) (2019)
- [A Note on Kernel Methods for Multiscale Systems with Critical Transitions](https://mlsanity.com/paper/1804.09415/) (2018)
- [Learning the Infinitesimal Generator of Stochastic Diffusion Processes](https://mlsanity.com/paper/2405.12940/) (2024)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [Diffusion map particle systems for generative modeling](https://mlsanity.com/paper/2304.00200/) (2023)
- [The Galerkin method beats Graph-Based Approaches for Spectral Algorithms](https://mlsanity.com/paper/2306.00742/) (2023)
- [From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2406.09028/) (2024)
- [Kernelized Diffusion maps](https://mlsanity.com/paper/2302.06757/) (2023)
- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning](https://mlsanity.com/paper/2311.12624/) (2023)
- [Learning solution operator of dynamical systems with diffusion maps kernel ridge regression](https://mlsanity.com/paper/2512.17203/) (2025)
- [Data-efficient Kernel Methods for Learning Hamiltonian Systems](https://mlsanity.com/paper/2509.17154/) (2025)
- [Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation](https://mlsanity.com/paper/2603.01077/) (2026)
- [Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics](https://mlsanity.com/paper/2603.06872/) (2026)

## Abstract (arXiv)

Selecting an appropriate kernel is a central challenge in kernel-based spectral methods. In 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.