Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces

TL;DR

This paper develops conformal and kNN-based uncertainty quantification methods for regression in metric spaces, providing finite-sample guarantees in homoscedastic cases and adaptive local methods for heteroscedastic settings, applicable to complex data types.

Contribution

It introduces a unified framework for uncertainty quantification in metric space regression, including conformal prediction with guarantees and adaptive local kNN methods without conformal calibration.

Findings

Conformal prediction achieves finite-sample coverage guarantees.

Local kNN method adapts to the geometry of nonlinear spaces.

Methods are scalable and applicable to personalized medicine data.

Abstract

This paper introduces a framework for uncertainty quantification in regression models defined in metric spaces. Leveraging a newly defined notion of homoscedasticity, we develop a conformal prediction algorithm that offers finite-sample coverage guarantees and fast convergence rates of the oracle estimator. In heteroscedastic settings, we forgo these non-asymptotic guarantees to gain statistical efficiency, proposing a local $k$--nearest--neighbor method without conformal calibration that is adaptive to the geometry of each particular nonlinear space. Both procedures work with any regression algorithm and are scalable to large data sets, allowing practitioners to plug in their preferred models and incorporate domain expertise. We prove consistency for the proposed estimators under minimal conditions. Finally, we demonstrate the practical utility of our approach in personalized--medicine applications involving random response objects such as probability distributions and graph Laplacians.