Sharp transport identification with unknown Gaussian covariance
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Abstract
Unknown Gaussian covariance creates an ambiguity between latent variation and noise. We quantify how covariance information controls this ambiguity in quadratic Wasserstein distance. On a fixed two-sided potential-curvature class, the sharp identification modulus is the latent-covariance uncertainty plus an inherited inverse logarithm of order 3/2. With n primary observations and m independent pure-noise observations, the sharp minimax rate replaces that uncertainty by its minimum with m⁻¹ᐟ². Retained independent-error pairs remove the covariance prior and recover the known-noise order. For rough log-concave inputs, a Gaussian-divisible reference gives linear covariance dependence and an exact compatible-law set. A bounded-variation heat estimate gives covariance dependence of order 3/4 for arbitrary smooth references, with sharp logarithmic testing under exact normalization. Finally, we allow unknown error shapes in a Wasserstein neighborhood of a Gaussian while preserving their covariance exactly. When the permitted noise covariance exceeds the unit latent scale, we prove a matching logarithmic shape-uncertainty floor and sharp primary-sample minimax risk. Positive Gaussian completion and covariance alignment transfer an established scalar-noise inverse theorem; exact-moment oscillations and convolution allocation supply the lower bounds.
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Research details
- Subjects
- Mathematics · Probability and Statistics
- Keywords
- Gaussian deconvolution, partial identification, covariance calibration, Wasserstein distance, minimax estimation
- Manuscript date
- 2026-10-31
- Reader reuse
- none
Scope
Quadratic Wasserstein identification and minimax estimation under bounded unknown Gaussian covariance, with latent-covariance priors, independent pure-noise calibration or retained independent-error pairs. Rough log-concave fixed-reference bounds use their stated normalization or Gaussian-divisibility assumptions. Exact-covariance Gaussian-shape neighborhoods have matching logarithmic ambiguity and primary-sample risk when S > 1; the general-reference covariance term is a sufficient bound of order ζ³ᐟ⁴.
Disclosures
OpenAI tools assisted with literature comparison, proof development, exposition and verification. The author assumes responsibility for all content. The author received no external research funding for this work. The author operates Luminary Oddities and its Research Archive, which hosts a preprint of this work.
Reproduction
The manuscript has 17 pages. The source package contains 5 files: main.tex, main.pdf, README.txt, CITATION.cff, 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.