Adaptive inference with random ellipsoids through Conformal Conditional Linear Expectation

TL;DR

This paper introduces new conformal prediction methods using ellipsoids derived from covariance analysis, providing theoretical guarantees and demonstrating improved volume efficiency over traditional methods in multivariate regression.

Contribution

It presents novel conformity scores based on residual covariance analysis, with explicit ellipsoidal prediction sets and proven asymptotic volume reduction under ellipticity assumptions.

Findings

Prediction ellipsoids have reduced volume compared to balls.

Theoretical guarantees ensure validity of the conformal sets.

Numerical studies confirm effectiveness on various distributions.

Abstract

We propose two new conformity scores for conformal prediction, in a general multivariate regression framework. The underlying score functions are based on a covariance analysis of the residuals and the input points. We give theoretical guarantees on the prediction sets, which consist in explicit ellipsoids. We study the asymptotic properties of the ellipsoids, and show that their volume is reduced compared to that of classic balls, under ellipticity assumptions. Finally, we illustrate the effectiveness of all our results on an in-depth numerical study, including heavy-tailed as well as non-elliptical distributions.