Group-theoretical vector space model

Abstract

This paper presents a group-theoretical vector space model (VSM) that extends the VSM with a group action on a vector space of the VSM. We use group and its representation theory to represent a dynamic transformation of information objects, in which each information object is represented by a vector in a vector space of the VSM. Several groups and their matrix representations are employed for representing different kinds of dynamic transformations of information objects used in the VSM. We provide concrete examples of how a dynamic transformation of information objects is performed and discuss algebraic properties involving certain dynamic transformations of information objects used in the VSM.

Keywords:

• vector space model
• group-theoretical vector space model
• group representation
• feature space
• information retrieval

2010 AMS Subject Classifications:

• 15A03
• 15A04
• 06B15
• 68Q55
• 68P20

Notes

1. Since a transformation F is often used for dimensionality reduction, it is distinguished from an (invertible) linear transformation in this paper. Note that an invertible linear transformation (i.e. isomorphism [15]) from a vector space V to itself serves as an element of the general linear group GL(V) (see the appendix).

2. By a slight abuse of notation, we use a document (respectively, a query) and its document vector (respectively, query vector) with the same notation in this paper. The distinction is clear from the context.

3. For a further consideration of a permutation of the basis vectors, consider a symmetric group S_n acting on a vector space $V={\mathbb{R}}^n$. Let $B=\{b_1,\dots ,b_n\}$ be a basis of V. Then, S_n acts on V by $g(\sum _i{c}_i{b}_i)=\sum _i{c}_i{b}_{g(i)}$ for g∈S_n, $c_i\in \mathbb{R}$, and $\sum _i{c}_i{b}_i\in V$. See [2,15] for further details.

4. See Appendix for Theorem A.1–A.4 and Lemma A.1–A.3.