Regression on the Gaussian Manifold: Bridging Bures-Wasserstein and Fisher-Rao Geometries for Inference on Normal-Valued Stochastic Processes

This article introduces an intrinsic geometric-statistical framework for regression with Gaussian-valued responses, treating multivariate normal distributions as elements of a finite-dimensional Riemannian manifold. The proposed methodology unifies two fundamental geometries on the Gaussian manifold: the Bures–Wasserstein geometry arising from optimal transport, and the Fisher–Rao information geometry induced by the statistical model. A family of mixed intrinsic loss functions is defined, yielding a non-parametric regression estimator based on weighted Fréchet means along Riemannian geodesics, while preserving positive-definiteness and invariance under congruence transformations. The paper establishes minimax-optimal convergence rates that depend explicitly on the intrinsic dimension of the Gaussian manifold, as well as a tangent-space central limit theorem characterizing the asymptotic distribution of the estimator. An explicit decomposition of the asymptotic covariance into mean and covariance components is derived, reflecting the product structure of the Gaussian manifold. In addition, a geometry-aware bootstrap procedure based on exponential-map perturbations of covariance operators is proposed, providing asymptotically valid inference directly on the space of Gaussian distributions. The framework connects non-parametric regression, information geometry, and optimal transport, offering a rigorous approach to inference for normal-valued stochastic processes.