Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces
This provides scalable uncertainty quantification tools for practitioners working with complex data in metric spaces, particularly in personalized medicine, though it builds incrementally on existing conformal and kNN approaches.
The paper tackles uncertainty quantification for regression models in metric spaces by introducing two algorithms: a conformal prediction method with finite-sample coverage guarantees for homoscedastic settings, and a kNN-based method for heteroscedastic settings that adapts to nonlinear geometries. The methods are demonstrated in personalized-medicine applications with random response objects like probability distributions and graph Laplacians.
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.