General pooling-bias conjecture for noncommuting heterogeneous Gaussian sources

Establish an analogous strictly positive, sample-size-independent Wasserstein bias between the pooled distribution and the Wasserstein barycenter whenever Gaussian source distributions differ simultaneously in their means and in non-commuting covariance matrices, and prove that this bias persists beyond the common-covariance and commuting-covariance special cases.

Background

The paper proves irreducible pooling bias in two special settings: heterogeneous means with a common covariance, and heterogeneous commuting covariances with a common mean. In both cases, the Wasserstein distance between the pooled distribution and the true Wasserstein barycenter remains strictly positive as sample sizes grow, whereas the empirical barycenter is consistent.

The unresolved case allows simultaneous mean and covariance heterogeneity with covariance matrices that are not jointly diagonalizable. In that setting, the pooled covariance and barycentric covariance generally share no common eigenbasis, so neither the Haar-twirling argument used for the common-covariance theorem nor the coordinatewise diagonal calculation used for the commuting-covariance proposition applies directly. The authors suggest that a perturbative bound involving covariance commutators could connect the commuting and fully noncommuting regimes.