Statistical inference for empirical heat-kernel Sinkhorn metrics
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Abstract
For a smooth positive density on a flat torus, the inverse weighted Laplacian defines energies of prescribed smooth tangent directions. We estimate a fixed finite matrix of these energies from independent observations, using symmetric heat-kernel Sinkhorn normalization without evaluating the density. The nonconstant spectral gap shrinks with the heat bandwidth, so statistical linearization must control the inverse uniformly in that bandwidth. We prove a scalar Sobolev Taylor estimate that supplies this control. Fixed multiband extrapolation removes the population bias and yields a joint, population-centered root-n Gaussian limit in every fixed dimension under explicit smoothness and bandwidth conditions. The influence function includes normalization, inversion and the changing integration measure. It gives a consistent covariance estimator, an exact nonparametric bootstrap and the pointwise information bound along fixed smooth density submodels. Separate observed error certificates transfer metric and covariance inference to finite numerical outputs. Their finite-arithmetic realization uses effective input enclosures and eventually informative spectral-gap enclosures. A circle example shows why freezing the normalized operator gives an incorrect asymptotic variance; exact rational fixtures check the numerical covariance error budgets. An explicit covariance rate also permits joint confidence cylinders with unknown limiting rank and certified numerical error. A circle example exhibits a noncompact second variation in a direction with zero first-order influence.
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Edition files
- Empirical_Sinkhorn_Metric_Inference.pdf193.5 KiB · pdf
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06d8948b68e6020f61f72e7584b3563a46549da98160581751092128228deb23 - Empirical_Sinkhorn_Metric_Publication_Package.zip231.2 KiB · package
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2151a65b3e64faae0ab2201b0610aa3d37aaf046dd530bc133b94e74a48075eb
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Research details
- Subjects
- Mathematics · Probability and Statistics
- Keywords
- Sinkhorn metric, diffusion maps, inverse weighted Laplacian, asymptotic linearity, semiparametric efficiency, observed stopping
- Manuscript date
- 2026-10-31
- Reader reuse
- none
Scope
Population-centered root-n inference for a fixed finite matrix of prescribed smooth-score inverse weighted-Laplacian energies on a fixed-dimensional flat torus, estimated by empirical heat-kernel Sinkhorn normalization with vanishing bandwidth and finite multiband bias correction. The results include complete influences, covariance estimation, exact bootstrap, smooth-submodel information bounds, certified numerical metric and covariance errors, unknown-rank confidence cylinders and a circle second-variation example.
Disclosures
ChatGPT Sol 6.1 assisted with 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 and received no external grant or sponsorship. The author is the founder of Luminary Oddities, which operates the Research Archive hosting the preprint. No other competing interests are declared. The finite rational fixtures and standard-library verification code are included in the accompanying source package. No empirical data from human participants were collected.
Reproduction
The manuscript has 22 pages. The source package contains 8 files: main.tex, singular_inference.tex, main.pdf, README.txt, CITATION.cff, check_covariance.py, check_covariance.json, MANIFEST.json. Extract the files into one directory and compile main.tex with a current LaTeX installation, for example: tectonic main.tex. Keep the included TeX files together. The paper is analytic and requires no empirical dataset. To reproduce the six exact rational covariance fixtures, run python check_covariance.py in a copy of the extracted directory. The script uses the Python standard library and writes check_covariance.json for comparison with the supplied output.