HyperVector and SuperHyperVector Spaces with Applications in Machine Learning: Feature, Support, and Relevance Vectors

Abstract

This paper introduces the concept of a SuperHyperVector Space, which extends classical vector spaces via the
SuperHyperstructure framework built on the ?
th iterated powerset. We first review how Hyperstructures and
SuperHyperstructures arise by applying the powerset and iterated powerset operations to a base set. We then
recall that a vector space consists of a set equipped with addition and scalar multiplication satisfying linearity
axioms, and that a hypervector space generalizes this structure by using a scalar hyperoperation that assigns to
each scalar–vector pair a nonempty subset of vectors while preserving distributivity and associativity. Building
on these ideas, we define SuperHyperVector Spaces by introducing a SuperHyperOperation on the iterated
powerset of the underlying group and briefly examine their fundamental properties and hierarchical modeling
potential. Furthermore, in the context of Machine Learning, we investigate extensions of the HyperVector
concept—including Feature Vector, Support Vector, and Relevance Vector—through the use of HyperVector
and SuperHyperVector representations.