Overview
This study investigates generalized Fréchet means defined on metric spaces possessing the Heine–Borel property. The research extends notions of centrality, such as the Fréchet mean, power means, geometric median, and the Chebyshev center, by replacing squared distance with a convex loss function, subject to a mild exponential-growth condition. The primary objective is to establish statistical properties for these generalized means, including a law of large numbers applicable beyond smooth manifolds and for varying choices of loss functions.
Research Context
Loss-based measures of central tendency are fundamental in summarizing data, particularly in nonlinear spaces. Traditional methods like the Fréchet mean are often defined using squared distance. This research expands this framework by considering a broader class of convex loss functions, referred to as $\phi$-means. The statistical utility of such central summaries necessitates a law of large numbers to ensure their consistency as data size increases. Current theoretical guarantees for these means are often restricted to contexts such as smooth manifolds or specific fixed loss functions.
Approach
The investigation utilized a theoretical approach rooted in a deterministic argmin principle combined with a Glivenko–Cantelli theorem for monotone classes. This methodological combination facilitated the analysis of generalized Fréchet means on metric spaces characterized by the Heine–Borel property. The definition of generalized Fréchet means involves replacing the squared distance with a convex loss function, subject to an exponential-growth condition. The framework aims to provide universal statistical guarantees for a range of loss functions and non-Euclidean spaces.
Findings
- The research demonstrates the existence and compactness of the population mean set for generalized Fréchet means.
- A sharp diameter bound for this population mean set was established.
- Almost-sure consistency of empirical $\phi$-means was proven.
- A uniform strong law was derived, applicable over compact classes of loss functions.
- For isotropic densities on Riemannian symmetric spaces, the population $\phi$-mean was identified for every strictly increasing loss where the objective function is finite, encompassing bounded robust losses.
- The framework's applicability was illustrated by examining examples on spheres and in the polyhedral space of ultrametric phylogenetic trees.
Why This Matters
The findings provide a foundational statistical framework for using loss-based notions of centrality in complex data spaces. By establishing uniform consistency and extending the law of large numbers to generalized Fréchet means on metric spaces with the Heine–Borel property, the research broadens the theoretical understanding and applicability of central tendency measures in non-Euclidean contexts. This theoretical underpinning supports the use of these summaries in various statistical analyses where data naturally reside in nonlinear spaces.