Adaptive kernel selection for Kernelized Diffusion Maps

Aboussaad, Othmane

doi:10.1016/j.physa.2026.131853

Adaptive kernel selection for Kernelized Diffusion Maps

2026-08-01

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DOI：10.1016/j.physa.2026.131853

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The selection of an appropriate kernel is a recurring challenge in kernel-based spectral methods. In Kernelized Diffusion Maps (KDM), the kernel fixes the RKHS geometry in which a diffusion-type operator is estimated, and hence the quality and stability of the recovered slow eigenfunctions. We formulate kernel choice as an unsupervised learning problem and develop a method that learns the KDM kernel by selection over multiple kernel families and bandwidths, optionally refined by a bounded variational outer loop that differentiates through the Cholesky-reduced eigenproblem. Our analysis isolates which objective to learn against: we prove that the common eigenvalue-sum cross-validation score is inconsistent for slow-subspace recovery when fast nuisance coordinates inflate the spectrum, and we establish an identifiability limit-no selector seeing only the static spectrum and the reproducibility of its top eigenspace can distinguish the slow subspace from a reproducible fast one. We resolve this in the dynamical setting that motivates the method: a VAMP-style time-lagged objective autonomously (kernel and lag, no labels) recovers the slow subspace on two real molecular-dynamics systems where the static spectral selectors fail. On alanine dipeptide it achieves SubR2 = 0.83 +/- 0.04-matching the best static oracle (spectral-gap CV, 0.83) without knowing the regime, far exceeding the eigenvalue-sum score (0.49), and beating a linear time-lagged baseline (TICA, 0.59); on chignolin folding it recovers the folding coordinate (0.53 vs. 0.22 for all static selectors). The nonlinear kernel is decisive on alanine and ties linear TICA on the near-onedimensional chignolin fold, so the system-independent gain is from the dynamical objective. The cross-validation pipeline achieves near-perfect recovery on low-dimensional Ornstein-Uhlenbeck processes (SubR2 up to 0.98, an order-of-magnitude subspace-error reduction) by automatically selecting non-Gaussian kernels, with about a 2 & times; error reduction in high-dimensional OU (d = 10-20) and more modest gains on the double-well and circle benchmarks. We further identify and analyze the structural failure modes of unconstrained gradient-based bandwidth optimization - sigma -> 0 collapse under orthonormality penalties and sigma -> infinity trivialization under eigenvalue-only losses - which dictate the method's bounded, anchored design. Finally, we frame kernel selection information-theoretically, through the minimum-description-length principle and a broader program bridging algorithmic information theory and machine learning, in which a good kernel yields a short, low-entropy description of the slow dynamics.

Keywords:

Kernelized diffusion maps

Kernel learning

Spectral estimation

Random Fourier features

Minimum description length

Algorithmic information theory

Kolmogorov entropy

Molecular dynamics

Slow collective variables

Adaptive kernel selection for Kernelized Diffusion Maps

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