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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, even with unrestricted unknown latent location. 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.