Recognizing weighted means in geodesic spaces

TL;DR

This paper investigates the problem of recognizing whether a point in geodesic spaces is an average or mean, focusing on nonpositively curved spaces and providing algorithms for efficient recognition.

Contribution

It introduces methods to identify averages in geodesic spaces, including a semidefinite programming algorithm for CAT(0) cubical complexes.

Findings

Recognition reduces to Euclidean projection in certain spaces

Efficient algorithms are developed for specific geodesic spaces

The approach applies to Hadamard manifolds and CAT(0) complexes

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.