Generalized Categorization Axioms

Categorization axioms have been proposed to axiomatizing clustering results, which offers a hint of bridging the difference between human recognition system and machine learning through an intuitive observation: an object should be assigned to its most similar category. However, categorization axioms cannot be generalized into a general machine learning system as categorization axioms become trivial when the number of categories becomes one. In order to generalize categorization axioms into general cases, categorization input and categorization output are reinterpreted by inner and outer category representation. According to the categorization reinterpretation, two category representation axioms are presented. Category representation axioms and categorization axioms can be combined into a generalized categorization axiomatic framework, which accurately delimit the theoretical categorization constraints and overcome the shortcoming of categorization axioms. The proposed axiomatic framework not only discuses categorization test issue but also reinterprets many results in machine learning in a unified way, such as dimensionality reduction, density estimation, regression, clustering and classification.