Brenier Isotonic Regression

Han Bao, Amirreza Eshraghi, Yutong Wang

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 When it comes to multivariate responses, IR is no longer applicable because monotonicity is not readily extendable.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.

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Code Dataset Url: https://github.com/levelfour/Brenier_Isotonic_Regression

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Submission Number: 409