Common-Center Geometry and Certified Radial Reconstruction for Energy-Form Full Conformal Regions

arXiv:2608.24964v1 Announce Type: cross Abstract: 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 when the candidate score is an empirical average of a loss convex in its first argument. Direct expansion of the leave-one-out scores shows that each training-point comparison for the energy-form score is exactly a pairwise-dissimilarity sublevel condition. Under symmetry, a constant diagonal, a diagonal lower bound, and attainment of the associated Frechet-type objective, every comparison region contains a common minimizer; when the comparison regions are convex, the nontrivial exact conformal region is therefore star-shaped about that same point. For power distances rho_eta(x,y)=|x-y|^eta, this deterministic geometry holds for etage1, while the conventional energy score is strictly proper for 0<eta<2. In the univariate eta=1 specialization, every nontrivial empirical-CRPS FullCP region is a nonempty closed interval, possibly mathbb R in the m=1 degeneracy. On the unconditional reconstruction range 1<eta<2 and mge2, explicit data-checkable derivative bounds yield Lipschitz control of the comparison-set radial exits and hence of the exact conformal radial function. These score-specific bounds permit existing directional root-search ideas and classical Lipschitz-extension machinery to yield certified inner and outer radial envelopes with width at most elta+2Lh_{mathcal U} and corresponding same-ray Hausdorff guarantees. An analytic two-dimensional example shows why retaining star-shaped but nonconvex geometry can matter. The resulting reconstruction perspective is intended for low-dimensional multivariate outputs rather than high-dimensional scaling or runtime improvement.

Source: arXiv cs.LG | 2026-08-27

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