Equiangular tight frames that contain regular simplices

• Published in: Linear algebra and its applications Vol. 555; pp. 98 - 138
• Main Authors: Fickus, Matthew, Jasper, John, King, Emily J., Mixon, Dustin G.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.10.2018; American Elsevier Company, Inc
• Subjects: Algorithms; Equiangular; Euclidean geometry; Euclidean space; Frame; Integer programming; Linear algebra; Packing problem; Subspaces; Tight; Frame; 42C15; Equiangular; Tight
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2018.06.004

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.