Scaling Dimension
This work addresses foundational gaps in Formal Concept Analysis, which is incremental as it builds on existing theory.
The paper tackles the problem of incomplete mathematical theory in Conceptual Scaling by extending basic notions and introducing the concept of scaling dimension, resulting in precise definitions, theoretical bounds, and initial results for special subclasses.
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.