Luminary Research Archive
Luminary Research ArchiveIndependent scholarship
Luminary Portal
PublicEdition 3

Statistical inference for empirical heat-kernel Sinkhorn metrics

Scott Schubr

Original PDF · First-page preview

Loading original first page…

Abstract

We estimate fixed-score inverse weighted-Laplacian energies of an unknown smooth strictly positive density on a flat torus using empirical symmetric heat-kernel Sinkhorn normalization. In every fixed dimension, fixed multiband bias correction yields a joint, population-centered root-n central limit theorem as the bandwidth vanishes. An explicit influence function accounts for normalization, inversion and the moving integration measure. We prove covariance consistency, validity of the exact nonparametric bootstrap, and pointwise efficiency along fixed smooth local density submodels. Separate observed certificates transfer metric and covariance inference to finite numerical output under effective input and eventually informative gap enclosures. Bandwidth conditions are conservative; finite termination gives no practical runtime or finite-sample coverage bound.

Read the full paper in the original PDF.

Edition files

Available editions

  1. Edition 5 · 2026-11-01 · public
  2. Edition 4 · 2026-10-31 · public
  3. Edition 3 · 2026-10-27 · public
  4. Edition 2 · 2026-10-26 · public
  5. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Probability and Statistics
Keywords
Sinkhorn metric, diffusion maps, inverse weighted Laplacian, asymptotic linearity, semiparametric efficiency, observed stopping
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Fixed smooth inverse weighted-Laplacian metric entries for a positive smooth torus density admit joint population-centered root-n Gaussian inference, consistent covariance and canonical-gradient efficiency under the stated conservative bandwidth and input-enclosure conditions.

Disclosures

ChatGPT Sol 6.1 assisted mathematical research, proof development, literature discovery, drafting and verification through workflows developed in the author's AI governance research. The author is responsible for all statements, proofs, attribution and presentation. The research was self-funded, without external grant or sponsorship. The author is the founder of Luminary Oddities, which operates this Archive; no other competing interests are declared.

Reproduction

The eight-member package includes the complete twenty-page manuscript, editable LaTeX, exact rational finite covariance controls and deterministic output, portable fixed-timestamp build helper, reader instructions, citation metadata and SHA-256 manifest. Complete implementation proofs are included in two appendices. Run check_covariance.py normally and under -O. These checks corroborate finite formulas; they are not a heat-kernel runtime or statistical coverage study.

References and related resources

  1. H. Lavenant, J. Luckhardt, G. Mordant, B. Schmitzer and L. Tamanini, The Riemannian geometry of Sinkhorn divergences (2026)
  2. C. L. Wormell and S. Reich, Spectral convergence of diffusion maps (2021)
  3. X. Cheng and B. Landa, Bi-stochastically normalized graph Laplacian (2024)
  4. N. Kessler, R. Magnet and J. Feydy, Sinkhorn Normalization of Diffusion Kernels (2026)
  5. Z. Goldfeld, K. Kato, G. Rioux and R. Sadhu, Limit Theorems for Entropic Optimal Transport Maps and the Sinkhorn Divergence, Electronic Journal of Statistics 18 (2024), 980–1041, doi:10.1214/24-EJS2217
  6. C. Li, N. García Trillos, H. Li and L. Suchan, Central limit theorems for the eigenvalues of graph Laplacians on data clouds (2025)
  7. T. Manole, S. Balakrishnan, J. Niles-Weed and L. Wasserman, Plugin Estimation of Smooth Optimal Transport Maps (2024)
  8. H. Kono, Central Limit Theorems for Optimal Transport Functionals (2026)
  9. G. Mordant and L. Xu, Convergence of the Sinkhorn Riemannian metric for finitely supported measures (2026)
  10. T. L. Saidi, F. Gunsilius and G. Mena, Second-order Sinkhorn Geometry and sqrt(n)-Inference (2026)
  11. S. Schubr, Sharp convergence of heat-kernel Sinkhorn metrics, edition 1 (2026)
  12. S. Schubr, Leading bias and positive correction of heat-kernel Sinkhorn metrics, edition 1 (2026)
  13. A. W. van der Vaart, Asymptotic Statistics, Cambridge University Press (1998)