  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.
