Geometry and Dress groups with non-symmetric cost functions

2025-05-31

AI Summary

Traditional symmetric metrics fail to model asymmetric data—such as causal networks, directed graphs, and time series—due to their inherent violation of symmetry. Method: This paper establishes the first systematic geometric theory of asymmetric cost spaces. It introduces the asymmetric Dress group, defines generalized curvature and betweenness-based order geometry tailored to asymmetry, and constructs a rigorous axiomatic framework grounded in abstract algebra, discrete geometry, and order theory. Contribution/Results: The work fundamentally relaxes the symmetry axiom of metric spaces, yielding an interpretable and computationally tractable geometric framework for asymmetric costs. It provides the first principled geometric foundation and analytical toolkit for directed structural data, enabling rigorous modeling, analysis, and inference on asymmetric relational structures. The theory bridges abstract mathematical formalism with practical applicability in domains requiring directional semantics, including causal discovery, temporal modeling, and network flow analysis.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

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.

Problem

• Develop theory for non-symmetric cost functions
• Study betweenness relations in non-symmetric datasets
• Introduce Dress groups and curvature in non-symmetric settings

Innovation

• Develops non-symmetric cost functions theory
• Utilizes betweenness relations for analysis
• Introduces Dress groups in non-symmetric context