Scale selection for geometric medians on product manifolds

2026-05-08

AI Summary

This work addresses the sensitivity of the geometric median on product manifolds to the relative scales of individual factor metrics, which leads to scale non-identifiability and optimization degeneracy. To circumvent boundary collapse caused by joint optimization, the paper introduces three novel strategies: sensitivity-aware pathways, scale-calibrated medians, and bounded equilibrium equations. Theoretical guarantees are established through functional central limit theorems, two-stage asymptotic expansions, bounded influence functions, and monotonic uniqueness estimates, ensuring consistency, robustness, and unit invariance of the proposed estimator. Empirical validation on both Euclidean spaces and the Bures–Wasserstein manifold demonstrates the effectiveness of the approach.

Abstract

Geometric medians on product manifolds are sensitive to the relative scaling of factor metrics because the median objective couples the factors rather than separating them. We study this scale-selection problem and first prove that naive joint minimization over location and scale is degenerate: the scale is driven to the boundary and the problem collapses to a marginal median, effectively discarding one factor. Thus relative scale is not identifiable from the raw median loss alone. We develop three alternatives to mitigate this issue. The first treats scale as indexing a sensitivity path and establishes uniform consistency, a functional central limit theorem, and a derivative-based sensitivity measure. The second constructs a robust scale-calibrated median using marginal radial median scales, yielding unit invariance, consistency, a two-step central limit theorem, and bounded influence. The third introduces a bounded balance equation for direct scale estimation, with monotonicity, uniqueness, joint asymptotic normality, and bounded influence. Simulations illustrate boundary collapse, sensitivity, unit invariance, and balanced estimation in Euclidean and Bures-Wasserstein settings.

Problem

• scale selection
• geometric median
• product manifolds
• relative scaling
• metric coupling

Innovation

• geometric median
• scale selection
• product manifolds
• robust estimation
• asymptotic theory