Sharp convergence of heat-kernel Sinkhorn metrics
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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.
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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
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.
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The complete publication package contains the exact seven-page paper, complete LaTeX source with embedded bibliography, reader/build instructions and SHA-256 checksums. The proofs are analytic.