{"ID":6023399,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-10T06:38:11.380144103Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05892","arxiv_id":"2607.05892","title":"On the convergence of graph Laplacians with a symmetric divergence","abstract":"When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\\mathcal{M}, g)$ of $\\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K \u003e 0$ such that $0 \\leq d_g(p, q)^2 - \\|p-q\\|^2 \\leq K d_g(p, q)^4$ for all $p, q \\in \\mathcal{M}$. We observe that more generally, when $\\mathcal{M}$ is equipped with a smooth symmetric divergence $D$ satisfying a non-degeneracy condition and $g$ is given by $g_p := \\frac{1}{2}\\mathrm{Hess}_p(D(p, \\cdot))$ for all $p \\in \\mathcal{M}$, there exists $K \u003e 0$ such that $\\left| D(p, q) - d_g(p, q)^2 \\right| \\leq K d_g(p, q)^4$ for all $p, q \\in \\mathcal{M}$. 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.","short_abstract":"When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\\mathcal{M}, g)$ of $\\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K \u003e 0$ such that $0 \\leq d_g(p, q)^2 - \\|p-q\\|^2 \\leq K d_g(p, q)^4$ for all $p, q \\in \\mathcal{M}...","url_abs":"https://arxiv.org/abs/2607.05892","url_pdf":"https://arxiv.org/pdf/2607.05892v1","authors":"[\"Liane Xu\"]","published":"2026-07-07T06:41:21Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}