Formal Concept Analysis

Definition

Formal concept analysis is a mathematical theory of concept hierarchies that builds on order theory; it can be seen as an unsupervised machine learning technique and is typically used as a method of knowledge representation. The approach takes an input binary relation (binary matrix) specifying a set of objects (rows) and a set of attributes for those objects (columns), finds the natural concepts described in the data, and then organizes the concepts in a partial order structure or Hasse diagram. Each concept in the final diagram is a pair of sets of objects and attributes that are maximally contained one in each other.

Theory

The above intuition can be formalized through a Galois connection as follows. Let R be the binary relation between a set of objects and a set of attributes, that is, $R \subseteq \mathcal{O}\times \mathcal{A}$. Two mappings $\alpha : \mathcal{O}\mapsto \mathcal{A}$ and $\beta : \mathcal{A}\mapsto \mathcal{O}$ are defined so that the operator α(O)...