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

Learning from multivariate distribution-valued data requires both averaging distributions and predicting them from covariates. Positive barycenters and Fr'echet regression impose different closure requirements. Regression weights can be negative, yet every fitted value must remain a probability law. We introduce the Hilbert mass-addressed parameterization (HiMAP), which uses balanced recursive median partitions to assign the same probability mass to each binary address across distributions. The common mass address yields two complementary representations. Q-HiMAP averages physical paths and defines a closed-form barycenter for nonnegative weights. Tree-logit HiMAP (TL-HiMAP) maps root geometry and relative split positions to Hilbert coordinates, where every finite affine combination decodes uniquely to a probability law. We establish exact mass coding, an encoder--decoder inverse, completeness, Wasserstein continuity, dense model coverage, and deterministic error decompositions for fixed-size particle outputs. The TL coordinates also give closed-form global and local Fr'echet regression and allow each response to be encoded once for repeated prediction. Experiments with certified barycenters, simulated distribution regression, and two NHANES survey cycles show accurate barycenter recovery, algebraically exact TL-coordinate composition with finite particle reconstruction error, competitive predictive loss, and efficient repeated prediction.