Quantitative Concept Analysis

TL;DR

This paper introduces a new framework called proximity sets (proxets) that extends Formal Concept Analysis to handle quantitative data, enabling extraction of quantified concepts and structural analysis of various FCA extensions.

Contribution

The paper proposes proxets, a categorical framework that unifies classical FCA and numeric extensions, allowing analysis of their universal properties and better utilization of quantitative information.

Findings

Proxets unify posets and metric spaces.

Quantified concepts can be extracted from numeric contexts.

Structural analysis guides the integration of FCA extensions.

Abstract

Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.