Adaptive Kernel Selection for Kernelized Diffusion Maps

2026-04-20

AI Summary

This work addresses the inaccuracy and instability of eigenfunction estimation in kernelized diffusion maps arising from the difficulty of kernel selection. To overcome this, two adaptive kernel selection strategies are proposed: first, a variational outer-loop optimization that jointly tunes continuous kernel parameters by maximizing eigenvalues, enforcing subspace orthogonality, and applying RKHS regularization; second, an unsupervised cross-validation procedure that combines eigenvalue-based criteria with random Fourier features to enable scalable selection of both kernel families and bandwidths. Theoretically, the study establishes Lipschitz dependence of the KDM operator on kernel weights, continuity of spectral projections, and residual control, and proves exponential consistency of the cross-validation selector over finite kernel dictionaries. Experiments demonstrate that the proposed methods substantially improve the accuracy, stability, and computational scalability of eigenfunction estimation.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• kernel selection
• Kernelized Diffusion Maps
• spectral methods
• RKHS estimation
• eigenfunctions

Innovation

• adaptive kernel selection
• kernelized diffusion maps
• differentiable eigenproblem
• unsupervised cross-validation
• spectral consistency