发表机构
School of Management, Fudan University（复旦大学管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析全共形预测区域的共中心几何，针对特定幂距离范围推导其星形等性质，提出可生成认证径向包络的方法，适用于低维多变量输出场景。
AI 中文摘要
本注研究由经验能量形式成对得分生成的全共形预测（FullCP）区域的几何性质。仅候选得分的凸性无法保证全共形区域连通，即使候选得分是对第一个参数为凸的损失的经验平均。直接展开留一法得分表明，能量形式得分的每个训练点比较恰好是成对不相似度的次水平条件。在对称性、常数对角线、对角线下界及关联弗雷歇型目标达成的条件下，每个比较区域包含一个共同极小值点；当比较区域为凸时，非平凡精确共形区域因此关于该同一点呈星形。对于幂距离ρ_β(x,y)=‖x−y‖^β，该确定性几何在β≥1时成立，而传统能量得分在0<β<2时严格恰当。在单变量β=1的特例中，每个非平凡经验连续秩概率得分（CRPS）全共形区域为非空闭区间，在m=1退化时可能为ℝ。在无条件重构范围1<β<2且m≥2时，显式可数据检验的导数界可实现比较集径向出口的利普希茨控制，进而实现精确共形径向函数的控制。这些特定得分的界允许现有方向根搜索思路和经典利普希茨延拓机制生成宽度至多为δ+2Lh_𝒰的认证内外径向包络，并对应同射线豪斯多夫保证。一个解析二维示例表明保留星形但非凸几何的重要性。所得重构视角旨在用于低维多变量输出，而非高维缩放或运行时改进。
英文摘要
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.
Comments29 pages, 3 figures. v2: tightened the radial certificate and added a staged numerical study of certificate tightness and robust nonconvexity diagnostics; conclusions and scope updated. Companion Lean 4 formalization and audited release: https://github.com/riasleyfung/fullcp-geometry-formal/tree/arxiv-v2-release
