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Leading bias and positive correction of heat-kernel Sinkhorn metrics

Scott Schubr

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Abstract

We identify the leading bias of the population heat-kernel Sinkhorn tensor at smooth strictly positive densities on a flat torus. Its second-order quadratic-form coefficient is minus one forty-eighth of the Wasserstein Hessian of Fisher information, with a fourth-order H6-to-L2 remainder uniform under every fixed order of density-parameter differentiation. An explicit density has bias directions of both signs. A compact same-bandwidth correction removes the bias, remains positive on the full tangent complement, and preserves the original completed form space and square-root range with equivalent norms. Its sharp Hr-to-L2 rates are h^(1+r/2) for 0 <= r <= 6; full L2 accuracy remains first order. Smooth finite-family metrics and intrinsic curvature inherit fourth-order accuracy. Separate computational appendices give certified metric values and normalization-inclusive approximation for prescribed analytic families close to uniform. The latter has logarithmic per-coordinate resolution under explicit coefficient and tail inputs.

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Available editions

  1. Edition 3 · 2026-11-02 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-22 · public

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

The exact leading population Sinkhorn tensor bias, a compact positive bias correction preserving its natural tangent topology, sharp Sobolev rates and prescribed-analytic-family finite approximations under the stated regularity and near-uniform hypotheses.

Disclosures

Author-posted preprint. OpenAI ChatGPT and Codex assisted mathematical exploration, literature research, derivation, drafting, proof checking and reproducibility code. The author assumes responsibility for all content.

Reproduction

The 22-page paper contains complete analytic proofs and two computational appendices. The ten-member package includes editable LaTeX, both interval scripts and full results, a portable build helper, citation metadata, reader instructions and checksums. Use mpmath 1.3.0 and reproduce the five continuum metric and three fixed-density normalization examples as documented, normally and under optimized Python.

References and related resources

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