Mind the Curvature: Correcting Structural Bias in Bures–Wasserstein Regression

Tangent-space linearization makes scalar-on-distribution regression tractable for multivariate Gaussian inputs, but its predictions depend on the reference distribution. In Bures–Wasserstein space, nonzero curvature can turn reference mismatch into a nonlinear projection error that persists even when the input distributions are known exactly. We characterize the part of this error that remains after refitting the linear predictor and use it to motivate learning the reference from prediction residuals. A local risk expansion at general positive definite references quantifies the joint effect of reference displacement and data spread, with a signal-dependent commutator coefficient at the identity. The result distinguishes positive geometric bias from zero leading coefficients and exact predictive equivalence. An alternating procedure fits the regression and updates the reference covariance, while an empirical-risk bound links reference selection to the population target. Exact-covariance experiments verify the predicted coefficients and scaling quantitatively. Repeated prediction experiments assess validation-based selection and separate geometric from estimation error. Wearable, molecular, and sensing applications show task-dependent prediction gains with the same linear readout.