On the convergence of graph Laplacians with a symmetric divergence

When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold (M, g) of ℝd, a key estimate for the geodesic distance dg is that there exists K > 0 such that 0 ≤ dg(p, q)2 - |p-q|2 ≤ K dg(p, q)4 for all p, q ∈ 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 gp := frac12Hessp(D(p, ·)) for all p ∈ M, there exists K > 0 such that left| D(p, q) - dg(p, q)2 right| ≤ K dg(p, q)4 for all p, q ∈ 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.