Scaling Dimension

TL;DR

This paper introduces the concept of scaling dimension in Formal Concept Analysis, providing precise definitions, exploring its properties, and establishing bounds related to the order dimension of concept lattices.

Contribution

It extends basic notions of conceptual scaling, introduces the new concept of scaling dimension, and analyzes its properties and bounds within concept lattices.

Findings

Defined the concept of scaling dimension.

Established theoretical bounds related to order dimension.

Analyzed special subclasses like ordinal and interordinal scaling dimensions.

Abstract

Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic definitions are in flux. Our contribution was triggered by the study of concept lattices for tree classifiers and the scaling methods used there. We extend some basic notions, give precise mathematical definitions for them and introduce the concept of scaling dimension. In addition to a detailed discussion of its properties, including an example, we show theoretical bounds related to the order dimension of concept lattices. We also study special subclasses, such as the ordinal and the interordinal scaling dimensions, and show for them first results and examples.