Leading bias and positive correction of heat-kernel Sinkhorn metrics
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Abstract
We identify the leading bias of the exact population heat-kernel Sinkhorn tensor on a flat torus and construct a compact, strictly positive correction with fourth-order smooth-input accuracy. The leading quadratic form is minus one forty-eighth of the Wasserstein Hessian of Fisher information, and an explicit nonconstant density exhibits both bias signs. The correction preserves the completed tangent form space and square-root range, with equivalent norms; full L² accuracy remains first order. An interval-certified nonconstant two-parameter example encloses continuum residual, differential-form and quadrature errors. For prescribed analytic density families sufficiently close to uniform, we construct fourth-order finite metric approximations, including normalization and every fixed parameter-derivative order. In each fixed dimension, per-coordinate Fourier resolution and iteration counts are logarithmic in inverse bandwidth; total modes and quadrature nodes grow as the dimension-th power of that logarithm. This is a resolution bound.
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- Leading_Bias_Positive_Correction_Sinkhorn_Metrics.pdf193.7 KiB · pdf
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e0b49d02e5cb7ab7ecc2fc81245f9a0a40a7ec15860e7ae288735447600bf11f - Leading_Bias_Positive_Correction_Sinkhorn_Publication_Package_20261021.zip283.5 KiB · package
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77871a0b8dbd006e336f35cadc200d89c2422dd8305f5636bebeeeb63cc729b2
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Research details
- Subjects
- Mathematics · Geometry and Optimal Transport
- Keywords
- Sinkhorn metrics, heat kernel, leading bias, Fisher information, positive correction, analytic densities, certified approximation
- Manuscript date
- 2026-10-21
- Reader reuse
- No additional permission granted
Scope
Exact population operators for smooth positive densities on a fixed-dimensional periodic torus. Fourth-order accuracy is H⁶ to L²; full L² convergence remains first order. The fixed prescribed-density theorem assumes the stated small analytic norm. This is a standalone follow-up to Sharp convergence of heat-kernel Sinkhorn metrics, edition 1.
Disclosures
AI tools assisted research, derivation, proof checking and manuscript preparation. This is an author-published preprint. Richardson and Simpson quadrature are credited classical methods.
Reproduction
The exact eight-member package contains the 22-page PDF, self-contained LaTeX source, two public Python reproducibility programs, two mathematical results files, README and checksums. Python examples require mpmath 1.3.0. The manufactured-density metric example and fixed prescribed-density normalization example have distinct scopes.