Brenier Isotonic Regression

2026-03-11

AI Summary

This work addresses the challenge of extending monotonicity constraints in multi-output regression, where traditional isotonic regression struggles to generalize. The authors propose a Brenier isotonic regression framework that, for the first time, integrates cyclic monotonicity with optimal transport theory. By introducing a Brenier potential function, they construct a regression mapping that satisfies cyclic monotonicity and establish its connection to the link function in generalized linear models. This approach overcomes the limitations of univariate monotonicity and effectively handles high-dimensional output settings. Experimental results demonstrate that the proposed framework significantly outperforms several classical baselines in tasks involving probability calibration and generalized linear modeling, exhibiting particularly strong robustness.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

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 When it comes to multi-output regression, the classical IR is no longer applicable because the monotonicity is not readily extendable.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.

Problem

• isotonic regression
• multi-output regression
• cyclically monotone
• shape-constrained regression
• monotonicity

Innovation

• Brenier isotonic regression
• cyclic monotonicity
• optimal transport
• convex potential
• multi-output regression