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

2026-08-25

AI Summary

研究通过能量形式的成对评分生成的全保形预测区域的几何特性，利用直接扩展留一法评分和特定边界条件下的方法解决了区域连通性问题。

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 $ρ_β(x,y)=\|x-y\|^β$, this deterministic geometry holds for $β\ge1$, while the conventional energy score is strictly proper for $0<β<2$. In the univariate $β=1$ 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 $1<β<2$ 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 $δ+2Lh_{\mathcal U}$ 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.

Problem

• Full Conformal Prediction
• Energy-Form Score
• Star-Shaped Region
• Geometric Reconstruction
• Multivariate Outputs

Innovation

• energy-form score
• full conformal prediction regions
• pairwise-dissimilarity sublevel condition
• star-shaped geometry
• Lipschitz control