Semantic Spaces

TL;DR

This paper explores the geometric and mathematical foundations of semantic spaces, particularly vector space models, to bridge natural language understanding in human brains and computational systems.

Contribution

It introduces a geometric framework using Grassmannians and projective spaces to model semantics and relates latent semantics to geometric flows, advancing the mathematical understanding of semantic spaces.

Findings

Geometric interpretation of latent semantics as flows on Grassmannians

Relation between vector space models and semantic spaces via projectability

Mathematical formulation of G"ardenfors' 'meeting of minds' concept

Abstract

Any natural language can be considered as a tool for producing large databases (consisting of texts, written, or discursive). This tool for its description in turn requires other large databases (dictionaries, grammars etc.). Nowadays, the notion of database is associated with computer processing and computer memory. However, a natural language resides also in human brains and functions in human communication, from interpersonal to intergenerational one. We discuss in this survey/research paper mathematical, in particular geometric, constructions, which help to bridge these two worlds. In particular, in this paper we consider the Vector Space Model of semantics based on frequency matrices, as used in Natural Language Processing. We investigate underlying geometries, formulated in terms of Grassmannians, projective spaces, and flag varieties. We formulate the relation between vector space models and semantic spaces based on semic axes in terms of projectability of subvarieties in Grassmannians and projective spaces. We interpret Latent Semantics as a geometric flow on Grassmannians. We also discuss how to formulate G\"ardenfors' notion of "meeting of minds" in our geometric setting.