HiMAP: Hilbert Mass-Aligned Parameterization for Multivariate Barycenters and Frećhet Regression

2026-03-03

AI Summary

This work addresses the computational complexity and ill-posedness of computing weighted Fréchet means in Wasserstein space for multivariate probability measures under affine weights—particularly those involving negative coefficients—which hinders the application of Fréchet regression. To overcome these limitations, the authors propose HiMAP, a novel method that constructs a distribution-invariant quantile map via recursive partitioning based on Hilbert curve-guided equal-probability conditional medians. This approach embeds probability measures into a Hilbert function space, enabling, for the first time, a parametric representation of the affine closure of multivariate measures and yielding well-defined barycenters even with negative weights. HiMAP supports closed-form affine averaging and explicit Fréchet regression, achieving computational efficiency significantly superior to optimal transport while maintaining comparable accuracy, as demonstrated in climate multivariate indicator experiments. Its theoretical convergence rate matches that of classical Wasserstein methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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 Frećhet 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 Frećhet 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.

Problem

• Wasserstein barycenter
• Fréchet regression
• multivariate probability measures
• affine averaging
• optimal transport

Innovation

• HiMAP
• Hilbert curve
• mass-aligned parameterization
• Fréchet regression
• Wasserstein barycenter