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Sharp convergence of heat-kernel Sinkhorn metrics

Scott Schubr

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Abstract

We prove sharp operator convergence of the periodic heat-kernel Sinkhorn metric at a smooth strictly positive density. The tensor hP_h(I-P_h^2)^{-1} converges to the inverse weighted Laplacian at rate O(h) in full L^2 operator norm and at rate O(h^{1+r/2}) from H^r to L^2, 0≤ r≤2. Each exponent is optimal already at constant density. Parameter-differentiated estimates give second-order convergence of smooth nondegenerate finite-family metrics, Christoffel symbols and intrinsic curvature. We also quantify the amplification of a population normalization residual and give an explicit projected scale approximation with O(h^2) tensor error. The results concern the stated bandwidth-dependent periodic cost and population model.

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Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-21 · public

Research details

Subjects
Mathematics · Geometry and Optimal Transport
Keywords
Sinkhorn divergence, heat kernel, weighted Laplacian, operator convergence, Sobolev estimates, metric curvature
Manuscript date
2026-10-21
Reader reuse
No additional permission granted

Scope

Sharp full L² and Sobolev convergence of periodic heat-kernel Sinkhorn tensors, parameter-differentiated intrinsic model geometry, and a population residual-to-tensor bound for smooth positive densities.

Disclosures

AI tools assisted research, derivation, proof checking and manuscript preparation. The author is responsible for the statements, attribution and presentation.

Reproduction

The seven-page paper contains complete analytic proofs. The six-member package includes editable LaTeX, a portable PDF build helper, citation metadata, reader instructions and checksums. No numerical experiment is a premise of the theorems.

References and related resources

  1. H. Lavenant, J. Luckhardt, G. Mordant, B. Schmitzer and L. Tamanini, The Riemannian geometry of Sinkhorn divergences, arXiv:2405.04987v2, 2025.
  2. G. Mordant and L. Xu, Convergence of the Sinkhorn Riemannian metric for finitely supported measures, arXiv:2609.14651v1, 2026.
  3. C. L. Wormell and S. Reich, Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation, arXiv:2006.02037v3, 2021.
  4. R. Yao, Y.-H. Kim and G. Schiebinger, Quantitative Stability of Many-Marginal Schrödinger Bridge, arXiv:2604.15191v1, 2026.
  5. X. Cheng and B. Landa, Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise, arXiv:2206.11386v3, 2024.
  6. T. Ichinose and H. Tamura, Results on Convergence in Norm of Exponential Product Formulas and Pointwise of the Corresponding Integral Kernels, arXiv:0905.3458, 2009.
  7. S. Schubr, First-Hit Stopping and Bias-Corrected Score Inference in Dimension Three, manuscript 2.1, public edition 2, 2026. Luminary Research Archive.
  8. S. Schubr, Leading bias and positive correction of heat-kernel Sinkhorn metrics, public edition 2, 2026. Luminary Research Archive.
  9. S. Schubr, Statistical inference for empirical heat-kernel Sinkhorn metrics, public edition 3, 2026. Luminary Research Archive.