Statistical inference for empirical heat-kernel Sinkhorn metrics
Loading original first page…
Abstract
We estimate a fixed matrix of inverse weighted-Laplacian energies for a smooth positive density on a flat torus, using empirical symmetric heat-kernel Sinkhorn normalization. As the heat bandwidth vanishes, the nonconstant spectral gap closes. A bandwidth-uniform scalar Sobolev Taylor bound controls this inverse, and fixed multiband extrapolation removes population bias. The resulting joint root-n Gaussian limit is centered at the limiting metric in every fixed dimension under explicit smoothness and bandwidth conditions. Its influence includes the changing normalization, inverse and integration measure, yielding covariance estimation, nonparametric bootstrap and the pointwise information bound along fixed smooth density submodels. Spectral selection permits joint inference with unknown limiting covariance rank. Separate observed metric and covariance error certificates transfer these conclusions to numerical outputs under effective input and informative gap enclosures. A circle example exhibits the variance contribution missed by freezing the normalized operator, and identifies the full second-variation spectrum of a trace with zero first-order influence.
Read the full paper in the original PDF.
Edition files
- Empirical_Sinkhorn_Metric_Inference.pdf192.9 KiB · pdf
SHA-256
ad66b383d2ce19937552fc011dda9e618693a1b1e7697c7850b6c405415ad771 - Empirical_Sinkhorn_Metric_Publication_Package.zip230.0 KiB · package
SHA-256
6cb8b53683e46f631d0936c4bc11eb267080c5e8c8047c51881a1f2618c7b2e8
Available editions
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 smooth-score inverse weighted-Laplacian energies on a fixed-dimensional flat torus, with vanishing heat bandwidth and fixed multiband correction. Complete influences, covariance estimation, bootstrap, pointwise information bounds and unknown-rank confidence cylinders use explicit regularity and bandwidth conditions. Numerical transfer uses separate metric and covariance certificates under effective input and informative gap enclosures; the circle example identifies a full continuum trace-Hessian spectrum.
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 all files into one directory and compile main.tex with a current LaTeX installation, for example: tectonic main.tex. Keep included TeX files together. The mathematical results are analytic. Run python check_covariance.py in a copy of the extracted directory to reproduce the six exact rational fixtures using the Python standard library; compare the generated check_covariance.json with the supplied output.