Liane Xu Proves Graph Laplacian Convergence with Symmetric Divergence

A new theoretical result from Liane Xu (arXiv:2607.05892) extends a foundational estimate in manifold learning to symmetric divergences, most notably the Sinkhorn divergence. In classical manifold learning, the geodesic distance d_g between points on a Riemannian submanifold satisfies a crucial quadratic approximation: |d_g^2 - Euclidean^2| ≤ K·d_g^4. Xu proves that the same quartic bound holds when the geodesic distance squared is replaced by a smooth symmetric divergence D, as long as D satisfies a non-degeneracy condition. The metric g is then derived from the Hessian of D. This narrows the gap between differential geometry and information geometry, providing theoretical justification for constructing graph Laplacians using divergences instead of distances.

Xu demonstrates the result's power with the Sinkhorn divergence — a regularized optimal transport distance defined on a family of probability measures parametrized by a manifold. This means graph Laplacian-based algorithms (e.g., spectral clustering, dimensionality reduction) can now be applied directly to probability distributions without first estimating geodesics. The 51-page paper includes 10 figures and rigorous analysis, opening the door to new manifold learning techniques on spaces of probability measures, with applications in generative models, computational anatomy, and geometric data analysis. The work bridges two fields — optimal transport and manifold learning — and provides the mathematical foundation for a new class of practical algorithms.