Statistical inference for empirical heat-kernel Sinkhorn metrics
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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. The stated sufficient bandwidth conditions govern the inference results.
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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
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
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. Existing population metric and bias papers remain separate and unchanged.
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.
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References and related resources
- C. L. Wormell and S. Reich, Spectral convergence of diffusion maps (2021)
- X. Cheng and B. Landa, Bi-stochastically normalized graph Laplacian (2024)
- N. Kessler, R. Magnet and J. Feydy, Sinkhorn Normalization of Diffusion Kernels (2026)
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- A. W. van der Vaart, Asymptotic Statistics, Cambridge University Press (1998)