Conformal inference for regression on Riemannian Manifolds
This work addresses the need for reliable uncertainty quantification in regression for non-Euclidean data, such as circular or covariance matrix data, which is incremental as it extends existing conformal inference methods to manifolds.
The paper tackles the problem of constructing prediction sets for regression when the response variable lies on a Riemannian manifold, extending conformal inference to this non-Euclidean context, and demonstrates its efficiency through simulations and real-world data analysis.
Regression on manifolds, and, more broadly, statistics on manifolds, has garnered significant importance in recent years due to the vast number of applications for non Euclidean data. Circular data is a classic example, but so is data in the space of covariance matrices, data on the Grassmannian manifold obtained as a result of principal component analysis, among many others. In this work we investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by $X$, lies in an Euclidean space. This extends the concepts delineated in \cite{waser14} to this novel context. Aligning with traditional principles in conformal inference, these prediction sets are distribution-free, indicating that no specific assumptions are imposed on the joint distribution of $(X,Y)$, and they maintain a non-parametric character. We prove the asymptotic almost sure convergence of the empirical version of these regions on the manifold to their population counterparts. The efficiency of this method is shown through a comprehensive simulation study and an analysis involving real-world data.