A short survey on almost orthogonal vectors in a few specific large dimensions

TL;DR

This survey explores the concept of almost orthogonal vectors in high-dimensional spaces, connecting mathematical theory, language model embeddings, and simulation methods to understand their properties and applications.

Contribution

It provides a comprehensive overview of almost orthogonal vectors across mathematical, computational, and AI contexts, highlighting recent developments and insights.

Findings

Almost orthogonal vectors are relevant in high-dimensional mathematics and coding theory.

Language models store concepts as almost orthogonal directions in embedding spaces.

Simulations can generate large sets of almost orthogonal vectors effectively.

Abstract

The concept of \emph{almost orthogonal vectors}, i.e.\ vectors whose cosine similarity is close to $0$, relates to topics both in pure mathematics and in coding theory under the guises of spherical packing and spherical codes. In recent years the rise of advanced language models in AI has created new interest in this concept as the models seem to store certain concepts as almost orthogonal directions in high-dimensional spaces. In this survey we represent some ideas regarding almost orthogonal vectors through three approaches: (1) the mathematical theory of almost orthogonality, (2) some observations from the embedding spaces of language models, and (3) generation of large sets of almost orthogonal vectors by simulations.