# Brenier Isotonic Regression

arXiv: https://arxiv.org/abs/2603.10452 · HTML: https://mlsanity.com/paper/2603.10452/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 65 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 24 of 24)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](https://mlsanity.com/paper/1910.12656/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Evaluating model calibration in classification](https://mlsanity.com/paper/1902.06977/) (2019)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Intra Order-preserving Functions for Calibration of Multi-Class Neural Networks](https://mlsanity.com/paper/2003.06820/) (2020)
- [Better Uncertainty Calibration via Proper Scores for Classification and Beyond](https://mlsanity.com/paper/2203.07835/) (2022)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Forecast Hedging and Calibration](https://mlsanity.com/paper/2210.07169/) (2022)
- [Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport](https://mlsanity.com/paper/1905.10812/) (2019)
- [Classifier Calibration with ROC-Regularized Isotonic Regression](https://mlsanity.com/paper/2311.12436/) (2023)
- [Uncoupled isotonic regression via minimum Wasserstein deconvolution](https://mlsanity.com/paper/1806.10648/) (2018)
- [Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation](https://mlsanity.com/paper/1903.01395/) (2019)
- [Fast Nonlinear Vector Quantile Regression](https://mlsanity.com/paper/2205.14977/) (2022)
- [Practical estimation of the optimal classification error with soft labels and calibration](https://mlsanity.com/paper/2505.20761/) (2025)
- [PAC-Bayes Analysis for Recalibration in Classification](https://mlsanity.com/paper/2406.06227/) (2024)
- [Proper losses regret at least 1/2-order](https://mlsanity.com/paper/2407.10417/) (2024)
- [Improving Multi-Class Calibration through Normalization-Aware Isotonic Techniques](https://mlsanity.com/paper/2512.09054/) (2025)
- [Least Squares Revisited: Scalable Approaches for Multi-class Prediction](https://mlsanity.com/paper/1310.1949/) (2013)
- [LegendreTron: Uprising Proper Multiclass Loss Learning](https://mlsanity.com/paper/2301.11695/) (2023)
- [Neural Optimal Transport Meets Multivariate Conformal Prediction](https://mlsanity.com/paper/2509.25444/) (2025)
- [Bilevel Optimization of the Kantorovich Problem and its Quadratic Regularization Part III: The Finite-Dimensional Case](https://mlsanity.com/paper/2406.08992/) (2024)

## Built on by (most cited first, 1 of 1)

- [CalArena: A Large-Scale Post-Hoc Calibration Benchmark](https://mlsanity.com/paper/2605.30188/) (2026)

## Abstract (arXiv)

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 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 Brenier isotonic regression. We demonstrate experiments with probability calibration and generalized linear models. In particular, IR outperforms many famous baselines in probability calibration robustly.