Scale selection for geometric medians on product manifolds

TL;DR

This paper investigates the scale selection problem for geometric medians on product manifolds, proposing three methods to address degeneracy and improve robustness, with theoretical guarantees and simulations.

Contribution

It introduces three novel approaches for scale calibration in geometric medians on product manifolds, overcoming degeneracy issues and ensuring consistency and robustness.

Findings

Naive joint minimization leads to boundary collapse and discards factors.

Three alternative methods achieve scale invariance and statistical consistency.

Simulations demonstrate effectiveness in Euclidean and Bures-Wasserstein contexts.

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.