On the convergence of graph Laplacians with a symmetric divergence

• URL: http://arxiv.org/abs/2607.05892v1
• Date: Tue, 07 Jul 2026 06:41:21 GMT
• Title: On the convergence of graph Laplacians with a symmetric divergence
• Abstract summary: 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 $p, q in mathcalM$.<n>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 para
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $K > 0$ 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 $K > 0$ 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.

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