On the convergence of graph Laplacians with a symmetric divergence

arXiv:2607.05892v1 Announce Type: cross Abstract: When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold (M, g) of R^d, a key estimate for the geodesic distance d_g is that there exists K > 0 such that 0 leq d_g(p, q)^2 - |p-q|^2 leq K d_g(p, q)^4 for all p, q in M. We observe that more generally, when M is equipped with a smooth symmetric divergence D satisfying a non-degeneracy condition and g is given by g_p := frac{1}{2}Hess_p(D(p, dot)) for all p in M, there exists K > 0 such that left| D(p, q) - d_g(p, q)^2 right| leq K d_g(p, q)^4 for all p, q in M. We demonstrate that this is sufficient for the pointwise convergence of graph Laplacians constructed with D and discuss examples where D is given by the Sinkhorn divergence on a family of probability measures parametrized by a manifold.

Source: arXiv cs.LG | 2026-07-08