Weighted Power Fréchet Means in Metric Spaces with Curvature Bounded Above

Abstract

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.