Research Article · Preprint Mathematics
Leading bias and positive correction 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 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, not a running-time or optimality claim.
Keywords Sinkhorn metrics · heat kernel · leading bias · Fisher information · positive correction · analytic densities · certified approximation
Scope of the work
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. No empirical complexity, faster solver, global corrected divergence, numerical curvature guarantee, dimension-uniform bound or fixed Euclidean-cost boundary result is asserted. This is a standalone follow-up to Sharp convergence of heat-kernel Sinkhorn metrics, edition 1.
Reproduction notes
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; the latter does not certify a full metric or parameter box.
The manuscript
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The complete manuscript, as published in this edition.
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Author disclosures
AI tools assisted research, derivation, proof checking and manuscript preparation. This is an author-published preprint; internal AI review is not external human peer review. Richardson and Simpson quadrature are credited classical methods.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Leading bias and positive correction of heat-kernel Sinkhorn metrics (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/leading-bias-positive-correction-heat-kernel-sinkhorn-metrics/v/1
BibTeX
@techreport{luminary_leading_bias_positive_correction_heat_kernel_sinkhorn_metrics_v1,
title = {Leading bias and positive correction 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/leading-bias-positive-correction-heat-kernel-sinkhorn-metrics/v/1}
}Loading discussion…