Ordinal spaces

TL;DR

This paper formalizes the concept of ordinal spaces in machine learning, exploring their properties, relationships with metric spaces, and conditions for embedding into Euclidean spaces, advancing the theoretical understanding of ordinal data analysis.

Contribution

It introduces the notion of ordinal spaces through an axiomatic approach and investigates their fundamental properties and connections to other mathematical structures.

Findings

Established axiomatic foundations for ordinal spaces

Analyzed conditions for isomorphism and embeddability

Explored topological and metric properties of ordinal spaces

Abstract

Ordinal data analysis is an interesting direction in machine learning. It mainly deals with data for which only the relationships `$<$', `$=$', `$>$' between pairs of points are known. We do an attempt of formalizing structures behind ordinal data analysis by introducing the notion of ordinal spaces on the base of a strict axiomatic approach. For these spaces we study general properties as isomorphism conditions, connections with metric spaces, embeddability in Euclidean spaces, topological properties etc.