# On the convergence of graph Laplacians with a symmetric divergence

arXiv: https://arxiv.org/abs/2607.05892 · HTML: https://mlsanity.com/paper/2607.05892/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 87 references that are in the map.
- First posted on arXiv 7 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds](https://mlsanity.com/paper/2506.00171/) (2025)
- [Gradient Flows of Potential Energies in the Geometry of Sinkhorn Divergences](https://mlsanity.com/paper/2511.14278/) (2025)
- [A kernel-based analysis of Laplacian Eigenmaps](https://mlsanity.com/paper/2402.16481/) (2024)
- [Laplace Learning in Wasserstein Space](https://mlsanity.com/paper/2511.13229/) (2025)

## Abstract (arXiv)

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.