Convergence of diffusion-geometry methods on non-manifold data

Establish convergence guarantees for the diffusion-geometry framework on general data geometries that are not manifolds. In particular, determine convergence (e.g., pointwise, spectral, and operator convergence) for the steps of eigenfunction estimation, carré du champ estimation, and Monte Carlo integration when the underlying space is a general probability space rather than a manifold.

Background

The paper outlines a three-step pipeline (eigenfunction estimation of the Markov chain, carré du champ estimation, and Monte Carlo integration) to build operators and solve weak formulations. These steps have established convergence properties primarily in manifold settings, often with fixed bandwidth kernels.

For general (non-manifold) data geometries, the authors state that the theoretical convergence of these components—and thus of the overall diffusion-geometry computations—has not been established.

References

First, the whole motivation for diffusion geometry is its use on general data sets, which may not lie on a manifold, but nothing is known about the above convergence in this setting.

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

However, extending these methods to surface-based skills remains an open issue, as surfaces encountered in real applications are rarely available as smooth manifolds.

— Robot Learning on Discrete Surfaces: Theory and Applications  (2610.01910 - Vedove et al., 1 Oct 2026) in Section 1, Introduction