Brenier Isotonic Regression

Han Bao, Amirreza Eshraghi, Yutong Wang
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:883-891, 2026.

Abstract

Isotonic regression (IR) is a shape-constrained regression to maintain a univariate fitting curve non-decreasing, which has numerous applications. When it comes to multivariate responses, IR is no longer applicable because monotonicity is not readily extendable. We consider a multi-output regression problem where a regression function is cyclically monotone. Roughly speaking, a cyclically monotone function is the gradient of some convex potential. Whereas enforcing cyclic monotonicity is apparently challenging, we leverage the fact that Kantorovich’s optimal transport (OT) always yields a cyclically monotone coupling. This naturally allows us to interpret a regression function and the convex potential as a link function in generalized linear models and Brenier’s potential in OT, respectively. We call this IR extension Brenier isotonic regression. We demonstrate applications to probability calibration and single-index models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-bao26a, title = { Brenier Isotonic Regression }, author = {Bao, Han and Eshraghi, Amirreza and Wang, Yutong}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {883--891}, 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/bao26a/bao26a.pdf}, url = {https://proceedings.mlr.press/v300/bao26a.html}, abstract = { Isotonic regression (IR) is a shape-constrained regression to maintain a univariate fitting curve non-decreasing, which has numerous applications. When it comes to multivariate responses, IR is no longer applicable because monotonicity is not readily extendable. We consider a multi-output regression problem where a regression function is cyclically monotone. Roughly speaking, a cyclically monotone function is the gradient of some convex potential. Whereas enforcing cyclic monotonicity is apparently challenging, we leverage the fact that Kantorovich’s optimal transport (OT) always yields a cyclically monotone coupling. This naturally allows us to interpret a regression function and the convex potential as a link function in generalized linear models and Brenier’s potential in OT, respectively. We call this IR extension Brenier isotonic regression. We demonstrate applications to probability calibration and single-index models. } }
Endnote
%0 Conference Paper %T Brenier Isotonic Regression %A Han Bao %A Amirreza Eshraghi %A Yutong Wang %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-bao26a %I PMLR %P 883--891 %U https://proceedings.mlr.press/v300/bao26a.html %V 300 %X Isotonic regression (IR) is a shape-constrained regression to maintain a univariate fitting curve non-decreasing, which has numerous applications. When it comes to multivariate responses, IR is no longer applicable because monotonicity is not readily extendable. We consider a multi-output regression problem where a regression function is cyclically monotone. Roughly speaking, a cyclically monotone function is the gradient of some convex potential. Whereas enforcing cyclic monotonicity is apparently challenging, we leverage the fact that Kantorovich’s optimal transport (OT) always yields a cyclically monotone coupling. This naturally allows us to interpret a regression function and the convex potential as a link function in generalized linear models and Brenier’s potential in OT, respectively. We call this IR extension Brenier isotonic regression. We demonstrate applications to probability calibration and single-index models.
APA
Bao, H., Eshraghi, A. & Wang, Y.. (2026). Brenier Isotonic Regression . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:883-891 Available from https://proceedings.mlr.press/v300/bao26a.html.

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