Generalized Hyperrectangular Prediction Sets in Multivariate Conformal Prediction

Nabil Alami, Souhaib Ben Taieb

In multivariate conformal prediction, many nonconformity scores yield prediction sets with complex geometry (e.g., ellipsoids or density level sets) that are difficult to interpret and visualize. Hyperrectangles offer a practical alternative by providing one interval per output. We extend this notion to Cartesian products of arbitrary one-dimensional sets, which we call generalized hyperrectangles. We introduce a rectangularization framework that turns any continuous multivariate nonconformity score, or conditional density, into one yielding generalized hyperrectangular prediction sets with finite-sample coverage. We establish key geometric properties of our rectangularization framework and provide useful intuitions on generalized hyperrectangles. We further establish a representation theorem showing that a broad class of hyperrectangular conformal methods based on max-aggregated coordinate-wise scores can be interpreted as rectangularizations of implicit conditional densities. Through synthetic and real-world experiments, we quantify the efficiency cost of rectangularization, while highlighting its benefits, especially for practical applications of uncertainty quantification.