Geometry and Dress groups with non-symmetric cost functions
Lukas Silvester Barth, Parvaneh Joharinad, Jürgen Jost, Walter Wenzel
TL;DR
This work generalizes metric-space geometry to non-symmetric relations by introducing non-symmetric cost functions and betweenness, and by developing a parallel theory of dresses groups (including a noncommutative version) and curvature. It extends Dress's hyperconvex hull concept to the non-symmetric setting via function pairs $(f,g)$, and builds a rich framework of tachistic/chronodesic curves, pretopologies, and path spaces, culminating in a directed notion of curvature and a non-symmetric tight span. By connecting cost spaces to Lawvere metric spaces, category enrichment, and Kolmogorov complexity, the paper provides a unified approach to analyzing directed dissimilarities, transport costs, and rewrite systems with potential applications in machine learning and computational linguistics. The contributions offer new algebraic and geometric tools for directed metrics, including generalized Dress groups, pretopological structures, and curvature measures, enabling directed analogues of classical concepts like hyperconvexity and geodesics with potential practical impact on non-symmetric data.
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.