Year: 2026
Type: preprint
Abstract: Isotonic regression (IR) is shape-constrained regression to maintain a univariate fitting curve non-decreasing, which has numerous applications including single-index models and probability calibration. When it comes to multi-output regression, the classical IR is no longer applicable because the monotonicity is not readily extendable. We consider a novel multi-output regression problem where a regression function is \emph{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 as an optimal solution. This perspective 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, and hence we call this IR extension \emph{Brenier isotonic regression}. We demonstrate experiments with probability calibration and generalized linear models. In particular, IR outperforms many famous baselines in probability calibration robustly.
Source: arXiv (Cornell University)
Authors: Bao, Han, Eshraghi, Amirreza, Wang, Yutong
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7135098523
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