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

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 Fréchet-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 $\rho_\beta(x,y)=\|x-y\|^\beta$, this deterministic geometry holds for $\beta\ge1$, while the conventional energy score is strictly proper for $0<\beta<2$. In the univariate $\beta=1$ 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 $1<\beta<2$ and $m\ge2$, 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 $\delta+2Lh_{\mathcal U}$ 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.

25 pages, 1 figure. Companion Lean 4 formalization and audited release: this https URL

Machine Learning (stat.ML); Machine Learning (cs.LG); Methodology (stat.ME)

arXiv:2608.24964 [stat.ML]

(or arXiv:2608.24964v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2608.24964 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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