HiMAP: Hilbert Mass-Aligned Parameterization for Multivariate Barycenters and Fre\'chet Regression

Many learning tasks represent responses as multivariate probability measures, requiring repeated computation of weighted barycenters in Wasserstein space. In multivariate settings, transport barycenters are often computationally demanding and, more importantly, are generally not well posed under the affine weight schemes inherent to global and local Fre\'chet regression, where weights sum to one but may be negative. We propose HiMAP, a Hilbert mass-aligned parameterization that endows multivariate measures with a distribution-invariant notion of quantile level. The construction recursively refines the domain through equiprobable conditional-median splits and follows a Hilbert curve ordering, so that a single scalar index consistently tracks cumulative probability mass across distributions. This yields an embedding into a Hilbert function space and induces a tractable discrepancy for distribution comparison and averaging. Crucially, the representation is closed under affine averaging, leading to a closed-form, well-posed barycenter and an explicit distribution-valued Fre\'chet regression estimator obtained by averaging HiMAP quantile maps. We establish consistency and a dimension-dependent polynomial convergence rate for HiMAP estimators under mild conditions, matching the classical rates for empirical convergence in multivariate Wasserstein geometry. Numerical experiments and a multivariate climate-indicator study demonstrate that HiMAP delivers barycenters and regression fits comparable to standard optimal-transport surrogates while achieving substantial speedups in schemes dominated by repeated barycenter evaluations.