Recognizing weighted means in geodesic spaces

TL;DR

This work tackles the problem of recognizing whether a given point in a geodesic metric space is a weighted mean (median or barycenter) of a finite set $A$, and shows that in Hadamard spaces the mean-set coincides with the $A$-minimal set, lying in the closed convex hull $\overline{\mathrm{conv}}(A)$. It develops a primal–dual optimization framework, analyzes the mean deficit via tangent-cone geometry, and provides existence results, unifying several mean notions. The authors specialize to Hadamard spaces and, in particular, CAT(0) cubical complexes, where mean-sets are computable and certifyable: in each maximal cell the mean reduces to Euclidean projections, while on cell boundaries semidefinite programming yields robust certificates and exact recognition through contraction-to-cell and SDP formulations. The results yield both theoretical insight into nonpositive curvature spaces and practical SDP-based algorithms for recognizing and certifying means in CAT(0) cubical complexes, with potential applications in geometric data analysis and phylogenetic metric geometry.

Abstract

Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally whether or not a given point is such an average, exactly or approximately. In nonpositively curved spaces, several averaging notions, including the usual weighted barycenter, produce the same "mean set". In such spaces, at points where the tangent cone is a Euclidean space, the recognition problem reduces to Euclidean projection onto a polytope. Hadamard manifolds comprise one example. Another consists of CAT(0) cubical complexes, at relative-interior points: the recognition problem is harder for general points, but we present an efficient semidefinite-programming-based algorithm.

Figures (6)

• Figure 1: A triangle and convex hull from three points in a geodesic metric space
• Figure 2: The mean set generated by three points in a geodesic metric space
• Figure 3: Heat maps for the mean deficit of three-point sets $\{a,b,c\}$ in CAT(0) square complexes, illustrating Examples \ref{['planar']} and \ref{['5squares']} respectively.
• Figure 4: The first panel shows a geodesic triangle with vertices $p,q,r$ in a CAT(0) cubical complex. The second panel shows the convex hull of $p,q,r$, which contains the three-dimensional simplex with vertices $p,0,s,r$.
• Figure 5: The first panel shows the mean set of three points, $p,q,r$, in a CAT(0) cubical complex in ${\mathbf R}^3$, a two-dimensional semi-algebraic surface. The second panel plots those randomly sampled points in the complex with mean deficit less than $\frac{1}{10}$.

Theorems & Definitions (47)

• Definition 2.1
• Proposition 2.4: Minimizers of the test function
• Definition 2.5
• Proposition 2.7
• Definition 2.8
• Proposition 2.9
• Lemma 2.10
• Proposition 2.11
• Proposition 2.12: Means in a Hilbert space
• Proposition 3.2: Weak minimax duality