Reproduce “Brenier Isotonic Regression” on OpenResearch

About this paper

Han Bao, Amirreza Eshraghi, Yutong Wang

Researchers introduce Brenier Isotonic Regression (BrenierIR), a nonparametric method that extends univariate isotonic regression to multivariate responses by enforcing cyclic monotonicity, a property linked to convex potentials and Optimal Transport theory. BrenierIR achieves competitive performance in multiclass probability calibration, often outperforming many baselines, and effectively learns cyclically monotone link functions in single-index models.

Key insights

• Cyclic monotonicity is a natural and theoretically sound generalization of univariate monotonicity for multi-output Generalized Linear Models where the inverse link function is the gradient of a convex potential.
• Optimal Transport theory, specifically Brenier's theorem, provides an indirect and computationally tractable method to enforce cyclic monotonicity, as optimal transport maps are inherently gradients of convex potentials.
• BrenierIR, through its barycentric and Laguerre maps, formally guarantees cyclic monotonicity for both training and test data, and reduces to standard univariate isotonic regression in the one-dimensional case.

Summary by alphaXiv.