Spectral convergence of variable-bandwidth diffusion kernels on manifolds

Determine whether eigenfunctions of variable-bandwidth diffusion kernels computed from manifold data converge to the corresponding Laplace–Beltrami eigenfunctions, i.e., establish spectral convergence of the Markov chain under variable bandwidth beyond pointwise convergence.

Background

Variable-bandwidth kernels are commonly used in practice due to their robustness on heterogeneous data. While pointwise convergence of the associated Markov chain has been established in some cases, spectral convergence (eigenfunction convergence) is critical for operator constructions and weak formulations.

The authors explicitly note that, even in manifold settings, current theory does not guarantee that eigenfunctions converge correctly for variable bandwidth kernels.

References

Second, in practice, diffusion methods are usually constructed with variable bandwidth kernels, and even for manifold data, we only have formal guarantees of pointwise convergence of the Markov chain, and do not currently know whether the eigenfunctions converge correctly.

— Computing Diffusion Geometry  (2602.06006 - Jones et al., 5 Feb 2026) in Section 3 (Frame theory and weak formulations), Subsection 'Towards overall convergence results'

We expect, however, that such a control is suboptimal with respect to the $L2(\mu_n)$-norm and that the correct rate is of order $h/ \sqrt{nv_\mu(h)}$. Obtaining such bounds likely requires additional structure on the metric-measure space (e.g., $X$ is a $d$-dimensional manifold). We leave this endeavor to future work.

— Spectral stability of empirical metric-measure Laplacians  (2608.23150 - Divol, 24 Aug 2026) in Section 1, subsection “Main theorem”