Close the theory–practice gap for diffusion geometry

Develop stronger theoretical results that match the empirically observed performance of diffusion geometry on general data geometries, including non‑manifold settings and practical choices such as variable bandwidth kernels and spectral cutoff regularization, thereby closing the gap between current theory and applications.

Background

The authors present robust empirical performance of their diffusion-geometry computations on non-manifold datasets and with practical regularizations, but note that theoretical guarantees lag behind practice.

They explicitly frame this disparity as an open problem, suggesting the potential for significantly stronger theoretical results than are currently available.

References

The lag of theory behind practice presents an open problem, where empirical performance (such as on the non-manifold examples in this paper) demonstrates that much stronger theoretical results could be obtained than those currently known.

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