Research Article · Preprint Mathematics
Sharp convergence of heat-kernel Sinkhorn metrics
- Primary subject
- Mathematics
- Subject subfields
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
We study the Sinkhorn metric associated with the periodic heat kernel at a smooth strictly positive probability density. If P_h is the symmetrically normalized heat operator, its metric tensor is hP_h(I-P_h^2)^(-1) on density-mean-zero functions. We prove convergence to the inverse weighted Laplacian in full L^2 operator norm at the optimal rate O(h). The sharper rate from H^r to L^2 is O(h^(1+r/2)), 0≤ r≤2; every exponent is optimal already at constant density. Parameter-differentiated estimates yield second-order convergence of smooth finite-dimensional pullback metrics and their intrinsic Christoffel symbols and curvature under nondegeneracy. A population normalization residual ρ gives projected tensor error O(h+ρ/h), with an obstruction at ρ≍ h. We also give an explicit projected scale approximation whose tensor differs from the exact tensor by O(h^2). The cost depends on the heat scale; fixed Euclidean squared-cost problems with boundaries are outside the scope.
Keywords Sinkhorn divergence · heat kernel · weighted Laplacian · operator convergence · Sobolev estimates · metric curvature
Scope of the work
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.
Reproduction notes
The paper has 7 pages. The editable-source archive contains 4 files. Compile main.tex; the paper supplies analytic proofs and an embedded bibliography.
The manuscript
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Author disclosures
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Cite this edition
Scott Schubr. (2026). Sharp convergence of heat-kernel Sinkhorn metrics (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-convergence-of-heat-kernel-sinkhorn-metrics/v/2
BibTeX
@techreport{luminary_sharp_convergence_of_heat_kernel_sinkhorn_metrics_v2,
title = {Sharp convergence of heat-kernel Sinkhorn metrics},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/sharp-convergence-of-heat-kernel-sinkhorn-metrics/v/2}
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