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.