ADOPT: Additive Optimal Transport Regression

2025-12-08

AI Summary

Regression with responses in general metric spaces—such as probability distributions, symmetric positive-definite (SPD) matrices, or spherical data—poses challenges for conventional additive models, which rely on vector-space structure and suffer from the curse of dimensionality in high-dimensional predictor settings. Method: We propose the first geodesically additive regression framework grounded in optimal geodesic transport, extending Wasserstein-optimal transport theory to arbitrary geodesic metric spaces. Without assuming linear structure, we define additivity via geodesic interpolation and combine it with geodesic function estimation to enable interpretable modeling of non-Euclidean responses. Contribution/Results: Empirical evaluation on fMRI-derived correlation matrices demonstrates that our method substantially mitigates the curse of dimensionality while achieving competitive predictive accuracy and providing clear, interpretable component-wise effects—marking a foundational advance in non-Euclidean additive regression.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Regression analysis for responses taking values in general metric spaces has received increasing attention, particularly for settings with Euclidean predictors $X in mathbb{R}^p$ and non-Euclidean responses $Y in ( mathcal{M}, d)$. While additive regression is a powerful tool for enhancing interpretability and mitigating the curse of dimensionality in the presence of multivariate predictors, its direct extension is hindered by the absence of vector space operations in general metric spaces. We propose a novel framework for additive optimal transport regression, which incorporates additive structure through optimal geodesic transports. A key idea is to extend the notion of optimal transports in Wasserstein spaces to general geodesic metric spaces. This unified approach accommodates a wide range of responses, including probability distributions, symmetric positive definite (SPD) matrices with various metrics and spherical data. The practical utility of the method is illustrated with correlation matrices derived from resting state fMRI brain imaging data.

Problem

• Extends additive regression to non-Euclidean response spaces.
• Uses optimal transport for interpretable, high-dimensional predictor modeling.
• Applies to diverse data like probability distributions and matrices.

Innovation

• Additive regression via optimal geodesic transports
• Extends optimal transport to general geodesic metric spaces
• Accommodates diverse responses like distributions and matrices