Equiangular tight frames that contain regular simplices

TL;DR

This paper explores equiangular tight frames containing regular simplices, characterizing their properties, developing algorithms to analyze their structure, and connecting them to optimal packings and fusion frames.

Contribution

It introduces a new characterization of ETFs with regular simplices, provides an algorithm for computing their binder, and links these structures to optimal subspace packings and fusion frames.

Findings

Characterization of ETFs containing regular simplices as those achieving a known compressed sensing bound.

Development of a new algorithm to compute the binder of such ETFs.

Identification of conditions under which ETFs form disjoint unions of regular simplices, leading to optimal subspace packings.

Abstract

An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an ETF, namely the set of all regular simplices that it contains. We provide a new algorithm for computing this binder in terms of products of entries of the ETF's Gram matrix. In certain circumstances, we show this binder can be used to produce a particularly elegant Naimark complement of the corresponding ETF. Other times, an ETF is a disjoint union of regular simplices, and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame. We conclude by considering the extent to which these ideas can be applied to numerous known constructions of ETFs, including harmonic ETFs.