Geometry and Dress groups with non-symmetric cost functions

TL;DR

This paper develops a systematic theory of non-symmetric cost functions in metric relations, introducing Dress groups and a notion of curvature to better model non-symmetric data sets.

Contribution

It introduces the concept of Dress groups and extends metric relations to non-symmetric cost functions, incorporating betweenness relations and curvature notions.

Findings

Established a framework for non-symmetric metric relations.

Defined Dress groups in the non-symmetric context.

Proposed a notion of curvature for non-symmetric spaces.

Abstract

A metric relation by definition is symmetric. Since many data sets are non-symmetric, in this paper we develop a systematic theory of non-symmetric cost functions. Betweenness relations play an important role. We also introduce the notion of a Dress group in the non-symmetric setting and indicate a notion of curvature.