Low frame coherence via zero-mean tensor embeddings

TL;DR

This paper introduces a novel embedding technique using higher degree polynomial maps to achieve optimal coherence in highly redundant real unit-norm frames, addressing limitations of equiangular arrangements.

Contribution

It extends existing embedding methods by employing quartic polynomial maps to embed highly redundant frames into high-dimensional simplices.

Findings

Higher degree polynomial embeddings improve frame coherence.

Quartic embedding maps enable better geometric arrangements for redundancy.

The method generalizes previous quadratic embeddings to higher degrees.

Abstract

This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other geometric optimality criteria need to be identified. To this end, we use an iteration of the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. We focus on the lowest degree case in which the embedding is quartic.