Sharp transport identification with unknown Gaussian covariance

Abstract

We determine how covariance information controls transport identification under unknown bounded anisotropic Gaussian noise. On a fixed two-sided potential-curvature class, the sharp quadratic Wasserstein modulus is the sum of the latent-covariance uncertainty and the inherited inverse logarithm of order 3/2. With n noisy observations and m independent pure-noise calibration observations, we prove the sharp minimax rate obtained by replacing that uncertainty with its minimum with m^-1/2. This identifies the calibration sample size needed to retain the known-noise rate. A Gaussian-divisible fixed reference permits rough log-concave inputs and has an explicit exact compatible-law set. Paired independent noisy repetitions supply a complementary calibration mechanism: they remove the covariance prior and yield sharp full-density estimation throughout the smooth class, together with fixed-reference testing against rough log-concave alternatives. Positive Gaussian completion and covariance alignment transfer an established scalar-noise inverse theorem to these experiments. Gaussian-allocation and moment-corrected oscillatory alternatives give the matching lower bounds.