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
- 1 · Permanent link
Overview
Abstract
We study the Sinkhorn metric associated with the periodic heat kernel at a smooth strictly positive probability density. We prove convergence to the inverse weighted Laplacian in full L² operator norm at the optimal rate O(h). The sharper rate from Hʳ to L² is O(h¹⁺ʳᐟ²), 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 comparable to h. We also give an explicit projected scale approximation whose tensor differs from the exact tensor by O(h²). 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
Smooth strictly positive periodic probability densities in each fixed dimension, with an h-dependent periodic heat-kernel cost. Sharp full L² and Sobolev tensor convergence rates; O(h²) intrinsic geometry of smooth nondegenerate finite-dimensional density families; population normalization sensitivity with explicit projection. Fixed Euclidean-cost boundary problems, vanishing densities, empirical guarantees, dimension-uniform bounds and ambient infinite-dimensional curvature are outside the scope.
Reproduction notes
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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 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-convergence-of-heat-kernel-sinkhorn-metrics/v/1
BibTeX
@techreport{luminary_sharp_convergence_of_heat_kernel_sinkhorn_metrics_v1,
title = {Sharp convergence of heat-kernel Sinkhorn metrics},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/sharp-convergence-of-heat-kernel-sinkhorn-metrics/v/1}
}Loading discussion…