Robust Estimation in metric spaces: Achieving Exponential Concentration with a Fréchet Median

TL;DR

This work extends robust Fréchet-median-based estimation to CAT($\kappa$) spaces, enabling exponential concentration under heavy tails. By introducing the Fréchet median of estimators (FMoE), the authors show how weakly concentrating estimators can be boosted to strong, exponential concentration with a bound of the form $\mathbb{P}[d(\hat{\theta}_{FMoE},\theta) > C_{\alpha}\epsilon] \le \exp(-k\psi(\alpha,p))$, where $C_{\alpha}$ and $\psi(\alpha,p)$ are explicitly defined. The framework is applied to Fréchet mean estimation in NPC spaces (via FMoM) and to covariance estimation on SPD manifolds under $d_{AI}$ and $d_{BW}$, with detailed concentration analyses and practical implementation guidance. Empirical results on metric trees and SPD examples corroborate the theoretical gains, illustrating substantial improvements in heavy-tailed settings and offering a tractable path for robust estimation in general metric spaces.

Abstract

There is growing interest in developing statistical estimators that achieve exponential concentration around a population target even when the data distribution has heavier than exponential tails. More recent activity has focused on extending such ideas beyond Euclidean spaces to Hilbert spaces and Riemannian manifolds. In this work, we show that such exponential concentration in presence of heavy tails can be achieved over a broader class of parameter spaces called CAT($κ$) spaces, a very general metric space equipped with the minimal essential geometric structure for our purpose, while being sufficiently broad to encompass most typical examples encountered in statistics and machine learning. The key technique is to develop and exploit a general concentration bound for the Fréchet median in CAT($κ$) spaces. We illustrate our theory through a number of examples, and provide empirical support through simulation studies.

Figures (7)

• Figure 1: Triangles in $M_{\kappa}^2$ spaces for different $\kappa$. Triangles in a CAT($\kappa$) space is thinner than triangles in $M_{\kappa}^2$. Left: Euclidean triangle ($\kappa = 0$). Middle: Spherical triangle ($\kappa > 0$). Right: Hyperbolic triangle ($\kappa < 0$).
• Figure 2: Left: A 5-legs spider tree. Right: One experiment result on the 5-legs spider tree. The origin denotes the population Frćhet mean, and red and green dot stand for inductive mean and Fréchet median of inductive means respectively.
• Figure 3: Histogram, mean, and 95% confidence interval for each experiment from 1000 simulations. Left: 5-legs spider. Middle: $(SPD, d_{AI})$. Right: $(SPD, d_{BW})$. All results indicate our method achieves much stronger concentration as well as much smaller mean squared errors.
• Figure 4: Possible configurations of the triangle $\triangle \widetilde{x}^* \widetilde{x}_j \widetilde{z}$ for $j = 1, \dots \left\lfloor(1-\alpha)k\right\rfloor + 1$. Clearly, the second case with the equality gives the tightest upper bound on $\sin(\widetilde{\gamma}_j)$, which is $1/C_{\alpha}$. This automatically yields the lower bound of the $\cos(\widetilde{\gamma}_j)$.
• Figure 5: The illustration of the Poincaré disk model. Poincaré disk model is defined by the intersection of the unit disk, e.g., $P_1, P_2$, and the projection map from the point $(t = -1, x = 0, y = 0)$ to the upper hyperboloid, e.g., $P_1', P_2'$. The distance between $P_1$ and $P_2$ is determined by the Euclidean length of the hyperbolic arc connecting their corresponding points $P_1'$ and $P_2'$.

Theorems & Definitions (38)

• Definition 2.1: Fréchet mean and median
• Remark 2.2: Existence and uniqueness of the Fréchet mean and median
• Lemma 3.1: Geometric discrepency near the Fréchet median
• Remark 3.2
• Theorem 3.3: Boosting a weak estimator
• Remark 3.4: $\kappa > 0$ case
• Remark 3.5: Time complexity of Algorithm \ref{['alg_moe']}
• Proposition 4.1
• proof
• Proposition 4.2