发表机构
Technical University of Munich; Potsdam Institute for Climate Impact Research（慕尼黑工业大学; 波茨坦气候影响研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为曲率上界度量空间中的加权幂 Fréchet 均值建立非渐近风险界，涵盖 Hadamard 和 CAT($\kappa$) 空间，并应用于 Dirichlet 过程 Fréchet 均值和局部常数 Fréchet 回归。
AI 中文摘要
我们为曲率上界的测地度量空间中的幂 Fréchet 均值建立了非渐近风险界。观测值构成加权、可能无限的独立随机变量序列，其分布和均值可能不同。我们处理三种情形。对于 Hadamard 空间中的 $2$-Fréchet 均值，我们获得一个尖锐的均方误差界，该界在 Hilbert 空间中变为恒等式。对于 $\kappa>0$ 的 CAT($\kappa$) 空间中的 $2$-Fréchet 均值，我们建立了方差和 Wasserstein 收缩不等式，其最优常数依赖于闭凸数据域的环半径 $\varrho$，并在最大范围 $\varrho<\pi/(2\sqrt{\kappa})$ 内获得均方误差界。对于 Hadamard 空间中的 $\alpha$-Fréchet 均值，$1<\alpha<2$，我们在加权 $\alpha$ 矩条件下推导出有限的 $L^\alpha$ 风险界，甚至允许总体混合的无限 $\alpha$ 矩。证明结合了方差、四重和收缩不等式以及留一稳定性技术。应用给出了 Dirichlet 过程 Fréchet 均值的先验和后验界，以及局部常数 Fréchet 回归的有限样本保证。回归结果不需要响应空间熵条件，并将早期工作中常见的密度平滑性假设替换为传输平滑性；对于 $\alpha<2$，即使响应具有无限方差，界仍然有限。
英文摘要
We establish non-asymptotic risk bounds for power Fréchet means in geodesic metric spaces with curvature bounded above. The observations form weighted, possibly infinite sequences of independent random variables whose laws and means may differ. We treat three settings. For $2$-Fréchet means in Hadamard spaces, we obtain a sharp mean squared error bound that becomes an identity in Hilbert spaces. For $2$-Fréchet means in CAT($κ$) spaces with $κ>0$, we establish variance and Wasserstein contraction inequalities with optimal constants depending on the circumradius $\varrho$ of the closed convex data domain, and obtain mean squared error bounds throughout the maximal range $\varrho<π/(2\sqrtκ)$. For $α$-Fréchet means in Hadamard spaces, $1<α<2$, we derive finite $L^α$ risk bounds under weighted $α$-moment conditions, allowing even an infinite $α$-moment of the population mixture. The proofs combine variance, quadruple, and contraction inequalities with a leave-one-out stability technique. Applications give prior and posterior bounds for Dirichlet-process Fréchet means and finite-sample guarantees for local constant Fréchet regression. The regression results require no response-space entropy condition and replace density smoothness assumptions common in earlier work with transport smoothness; for $α<2$, the bounds remain finite even when the responses have infinite variance.
