This note studies the geometry of full conformal prediction (FullCP) regions generated by an empirical energy-form pairwise score. Candidate-score convexity alone does not guarantee connected FullCP regions, even for empirical averages of losses convex in the candidate argument. For the energy-form score, each leave-one-out training comparison reduces exactly to a pairwise-dissimilarity sublevel condition. Under symmetry, a constant diagonal, a diagonal lower bound, and attainment of the associated Fréchet-type objective, all comparison regions share a minimizer; if they are convex, every nontrivial exact conformal region is star-shaped about that point. For power distances $ρ_β(x,y)=\|x-y\|^β$, this geometry holds for $β\ge1$, while the conventional energy score is strictly proper for $0<β<2$. For $d=1,β=1$, every nontrivial empirical-CRPS FullCP region is a nonempty closed interval (possibly $\mathbb R$ when $m=1$). For $1<β<2$ and $m\ge2$, explicit data-checkable derivative bounds give Lipschitz control of the radial exits and exact conformal radial function. Combined with directional root search and classical Lipschitz extensions, they yield certified inner and outer radial envelopes of width at most $δ+2\widetilde Lh_{\mathcal U}$ and same-ray Hausdorff guarantees. An analytic two-dimensional example shows why preserving star-shaped but nonconvex geometry can matter. A staged two-dimensional study finds modest but systematic tightening of the generic certificate and frequent robust nonconvexity witnesses, with detected normalized radial departures typically small. The method is intended for low-dimensional multivariate outputs rather than high-dimensional scaling or runtime improvement.
Engression (Shen and Meinshausen, 2024) learns a conditional distribution by fitting a generative model $Y = f(X,\varepsilon)$ under the energy score, a strictly proper scoring rule. We provide a theoretical error analysis of engression implemented with deep neural networks. We decompose the excess risk into three components: the approximation error, the stochastic error, and the Monte Carlo error. Based on this decomposition, we establish convergence rates under the assumption that the target conditional generator admits a compositional smoothness structure.
Circular data, representing angles or directions, are frequently encountered in computer vision, biology, geology, and meteorology. Traditional regression targets the conditional mean, which is often geometrically misleading for circular responses under multimodal, skewed, or asymmetric data structures. To address these limitations, a lightweight deep generative framework, namely ANGLE, is introduced for non-parametric distributional regression on the circle. The full conditional distribution of an angular response, given Euclidean and circular covariates, is learned through a generative map optimized via a generalized circular energy score (GCES) loss. Desirable theoretical properties, including the strict propriety of the loss and the rotational equivariance of the estimators, are established. Furthermore, both pre- and post-additive noise models are accommodated. A unified toolbox is provided for advancing previously underexplored challenges in circular statistics: extrapolation, sufficient dimension reduction, and conditional distribution equality testing. The framework's efficacy is demonstrated through extensive simulations and real-world applications. Specifically, the proposal is utilized for object pose estimation from imagery and wind direction prediction, which are integral to surveillance, autonomous vehicles, and energy systems, respectively. Superior predictive performance and robust uncertainty quantification of the proposed method in these tasks are revealed.
We propose Emputation, a deep generative framework for learning imputation models. Emputation targets the extrapolation distribution of missing variables given observed variables, and training is guided by specific missingness assumptions that guarantee identification of the target distribution. The training objective, called the emputation risk, is an energy-score-based risk in which the identification assumption determines how observed entries are masked and which observations contribute to training. The resulting framework enables direct conditional sampling for multiple imputation. We show that the population minimizer of the emputation risk recovers the target extrapolation distribution under a broad class of identification assumptions, including several missing-not-at-random assumptions. Simulations show strong performance under both pointwise and distributional evaluation metrics, and an application to an Alzheimer's disease dataset demonstrates its practical value.