ADOPT: Additive Optimal Transport Regression

Wookyeong Song, Hans-Georg Müller
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:1558-1566, 2026.

Abstract

Regression models for responses $Y$ taking values in general metric spaces $(\mathcal{M}, d)$, with Euclidean predictors $X \in \mathbb{R}^p,$ has attracted growing interest in recent years. 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-song26c, title = { ADOPT: Additive Optimal Transport Regression }, author = {Song, Wookyeong and M{\"u}ller, Hans-Georg}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {1558--1566}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/song26c/song26c.pdf}, url = {https://proceedings.mlr.press/v300/song26c.html}, abstract = { Regression models for responses $Y$ taking values in general metric spaces $(\mathcal{M}, d)$, with Euclidean predictors $X \in \mathbb{R}^p,$ has attracted growing interest in recent years. 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. } }
Endnote
%0 Conference Paper %T ADOPT: Additive Optimal Transport Regression %A Wookyeong Song %A Hans-Georg Müller %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-song26c %I PMLR %P 1558--1566 %U https://proceedings.mlr.press/v300/song26c.html %V 300 %X Regression models for responses $Y$ taking values in general metric spaces $(\mathcal{M}, d)$, with Euclidean predictors $X \in \mathbb{R}^p,$ has attracted growing interest in recent years. 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.
APA
Song, W. & Müller, H.. (2026). ADOPT: Additive Optimal Transport Regression . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:1558-1566 Available from https://proceedings.mlr.press/v300/song26c.html.

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