Super-Level-Set Regression: Conditional Quantiles via Volume Minimization

TL;DR

This paper introduces super-level-set regression (SLS), a novel geometric framework for directly optimizing conditional quantile regions in multivariate regression, bypassing density estimation.

Contribution

SLS provides a new direct geometric optimization approach for conditional quantile regions, capturing complex structures without density estimation.

Findings

SLS effectively captures complex, multimodal, and disjoint conditional structures.

The method bypasses the need for explicit density estimation, reducing sensitivity to errors.

SLS offers a new perspective on multivariate conditional quantile regression.

Abstract

Constructing minimum-volume prediction regions that satisfy conditional coverage is a fundamental challenge in multivariate regression. Standard approaches rely on explicitly estimating the full conditional density and subsequently thresholding it. This two-step plug-in process is notoriously difficult, sensitive to estimation errors, and computationally expensive. One would like to instead optimize the region directly. Formulating a direct solution is challenging, however, because it requires minimizing a volume objective that is coupled with the conditional quantiles of the model's own estimation error. In this work, we address this challenge. We introduce super-level-set regression (SLS), a novel mathematical framework that successfully resolves this implicit coupling, allowing us to directly parameterize and optimize the geometric boundaries of the target conditional level sets. By bypassing full distribution estimation and leveraging flexible volume-preserving frontier functions, our approach natively captures complex, multimodal, and disjoint conditional structures end-to-end. Ultimately, SLS offers a new perspective on multivariate conditional quantile regression, replacing the restrictive assumptions of density-first methods with a direct geometric optimization strategy.