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Research Article · Preprint Mathematics

Statistical inference for empirical heat-kernel Sinkhorn metrics

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Probability and Statistics
Primary subject
Mathematics
Subject subfield
Probability and Statistics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

We estimate fixed smooth pullback entries of the inverse weighted-Laplacian metric of an unknown smooth strictly positive density on a flat torus. The estimator uses symmetric heat-kernel Sinkhorn normalization of the sample and prescribed score functions, without density evaluations. For every fixed dimension we establish a joint, population-centered root-n central limit theorem as the heat bandwidth tends to zero. The proof gives a signed-measure Taylor expansion on one Sobolev space, an explicit influence function including normalization and inverse response, and a consistent empirical covariance estimator using exact solves. A scalar Hessian identity with the moving integration measure improves the sufficient remainder budget. The population tensor has even finite asymptotic jets; fixed multiband extrapolation removes enough bias to retain the same limiting covariance. The influence is the canonical gradient, and the estimator attains the pointwise information bound under smooth local density submodels. Finally, an observed scaling, spectral-gap and inverse-residual certificate produces an actual finite first-hit output with negligible root-n metric error. Its certified-arithmetic realization requires effective convergent input enclosures. Bandwidth conditions are conservative; no practical runtime, optimal smoothing rate or numerically certified covariance algorithm is asserted.

Keywords Sinkhorn metric · diffusion maps · inverse weighted Laplacian · asymptotic linearity · semiparametric efficiency · observed stopping

Scope of the work

The results concern an unknown fixed smooth strictly positive density on a fixed-dimensional torus and a fixed finite collection of smooth prescribed exponential-tilt scores under periodic heat normalization. They establish population-centered joint inference, conservative bandwidth conditions, even-band bias cancellation, pointwise efficiency in smooth local density submodels and an observed metric-error numerical-equivalence certificate under effective-enclosure assumptions. They do not establish optimal bandwidth selection, practical runtime, rough or vanishing densities, parameter-uniform or growing-dimensional inference, curvature inference, fixed squared-cost transport results or a numerically certified covariance implementation. Existing population metric and bias papers remain separate and unchanged.

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Author disclosures

AI systems assisted mathematical development, literature discovery, proof checking and manuscript preparation, as disclosed in the paper. The author is responsible for the statements, attribution and presentation. The final analytic and literature reviews are internal AI reviews, not external human refereeing or formal proof verification. Novelty remains provisional within the recorded primary-source comparisons and access limits.

References and related publications

  1. C. L. Wormell and S. Reich, Spectral convergence of diffusion maps (2021)
  2. X. Cheng and B. Landa, Bi-stochastically normalized graph Laplacian (2024)
  3. N. Kessler, R. Magnet and J. Feydy, Sinkhorn Normalization of Diffusion Kernels (2026)
  4. Z. Goldfeld, K. Kato, G. Rioux and R. Sadhu, Limit Theorems for Entropic Optimal Transport Maps and the Sinkhorn Divergence (2024)
  5. C. Li, N. García Trillos, H. Li and L. Suchan, Central limit theorems for the eigenvalues of graph Laplacians on data clouds (2025)
  6. T. Manole, S. Balakrishnan, J. Niles-Weed and L. Wasserman, Plugin Estimation of Smooth Optimal Transport Maps (2024)
  7. H. Kono, Central Limit Theorems for Optimal Transport Functionals (2026)
  8. H. Lavenant, J. Luckhardt, G. Mordant, B. Schmitzer and L. Tamanini, The Riemannian geometry of Sinkhorn divergences (2026)
  9. G. Mordant and L. Xu, Convergence of the Sinkhorn Riemannian metric for finitely supported measures (2026)
  10. S. Schubr, Sharp convergence of heat-kernel Sinkhorn metrics, edition 1 (2026)
  11. S. Schubr, Leading bias and positive correction of heat-kernel Sinkhorn metrics, edition 1 (2026)
  12. A. W. van der Vaart, Asymptotic Statistics, Cambridge University Press (1998)

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Statistical inference for empirical heat-kernel Sinkhorn metrics (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/statistical-inference-empirical-heat-kernel-sinkhorn-metrics/v/1

BibTeX
@techreport{luminary_statistical_inference_empirical_heat_kernel_sinkhorn_metrics_v1,
  title = {Statistical inference for empirical heat-kernel Sinkhorn metrics},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/statistical-inference-empirical-heat-kernel-sinkhorn-metrics/v/1}
}

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