Variance Inequalities for Transformed Fréchet Means in Hadamard Spaces

TL;DR

This work develops a unified theory of transformed Fréchet means in Hadamard spaces by introducing $\tau$-Fréchet means for $\tau \in \mathcal{S}$, which encompasses the Fréchet mean, Fréchet median, and robust statistics like the Huber loss. It proves a general variance-inequality framework, including a central Hadamard-space bound $\mathbb{E}[\tau(d(Y,q)) - \tau(d(Y,m))] \geq \tfrac{1}{2} d(q,m)^2 \mathbb{E}[\tau^{\oplus}(\max(d(Ym),d(Yq)))]$ under mild moment conditions, and shows quadratic growth near minimizers when $\tau''>0$. The paper also provides a dedicated median analysis with near-median and geodesic-concentration results, delivering uniqueness criteria and variance bounds for the Fréchet median in Hadamard spaces. By combining $\mathcal{G}$-convexity, quadratic lower bounds, and Hadamard geometry, the authors obtain estimation- and approximation-ready results with minimal moment assumptions, broadening robust central-tendency inference in nonpositively curved metric spaces.

Abstract

The Fréchet mean (or barycenter) generalizes the expectation of a random variable to metric spaces by minimizing the expected squared distance to the random variable. Similarly, the median can be generalized by its property of minimizing the expected absolute distance. We consider the class of transformed Fréchet means with nondecreasing, convex transformations that have a concave derivative. This class includes the Fréchet median, the Fréchet mean, the Huber loss-induced Fréchet mean, and other statistics related to robust statistics in metric spaces. We study variance inequalities for these transformed Fréchet means. These inequalities describe how the expected transformed distance grows when moving away from a minimizer, i.e., from a transformed Fréchet mean. Variance inequalities are useful in the theory of estimation and numerical approximation of transformed Fréchet means. Our focus is on variance inequalities in Hadamard spaces - metric spaces with globally nonpositive curvature. Notably, some results are new also for Euclidean spaces. Additionally, we are able to characterize uniqueness of transformed Fréchet means, in particular of the Fréchet median.

Figures (4)

• Figure 1: Different transformation functions $\tau\in\mathcal{S}$ and their derivatives. The functions shown are $\tau_\alpha(x) = \alpha^{-1}x^\alpha$ for $\alpha \in \{1, 3/2, 2\}$ as well as the Huber and pseudo-Huber loss functions with threshold $\delta=1$, see \ref{['eq:huber']} and \ref{['eq:pseudohuber']}, respectively.
• Figure 2: The Fréchet median in the stick figure space. The stick figure shown here is to be taken as a subset of $\mathbb{R}^2$ with its intrinsic distance. It is a Hadamard space: Convex subsets of the Euclidean plane are Hadamard spaces. Moreover, gluing Hadamard spaces together yields a new Hadamard space according to Reshetnyak’s Gluing Theorem sturm03. Thus, the stick figure, which is obtained from gluing together a disk and some line segments is Hadamard. For (a): Let $\gamma\in\Gamma_1$ be the geodesic starting at the stick figures neck such that its image is the purple torso. Then the red head is the left set $L(\gamma)$ and the green legs make up the right set $R(\gamma)$. The arms do not belong to either set. In (b) -- (f) the orange color depicts a distribution where small circles with a number on the side indicate a point mass with that probability and the orange half circle in (e) is to be taken as a uniform distribution on the colored area of one half of the total mass. In (f) the probability mass in the legs and in the head each make up one half in total. The Fréchet median set is shown in purple. It is either a geodesic segment of positive length in (c) -- (e), (g) or a single point in (b), (f).
• Figure 3: Visualization of the bowtie complement $A(m, q, \eta)$ in $\mathbb{R}^2$. The Euclidean plane is depicted as a gray area. The image of a geodesic $\gamma\colon\mathbb{R}\to\mathbb{R}^2$ with $\gamma(0) = m$ and $\gamma(\overline{qm}) = q$ is shown in black. The sets $\{y\in\mathbb{R}^2 \mid \overline{y\gamma}^\oplus(0) > 1 - \eta^2\}$ and $\{y\in\mathbb{R}^2 \mid \overline{y\gamma}^\oplus(\overline{qm}) > 1 - \eta^2\}$ are depicted as purple and orange dotted areas, respectively. The set $A(m, q, \eta)$ is the gray area without the dotted areas.
• Figure 4: Construction of $A_r$ (green area) in the proof of \ref{['cor:varinequ:median']} illustrated in $\mathbb{R}^2$. Also confer \ref{['fig:medianCutOut']}.

Theorems & Definitions (111)

• Definition 1.1
• Proposition 1.2: sturm03
• Definition 2.1
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Remark 2.6
• Proposition 3.1
• proof
• Proposition 3.2