Equiangular Tight Frames That Contain Regular Simplices
Document Type
Article
Publication Date
10-15-2018
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.
Source Publication
Linear Algebra and its Applications (ISSN 0024-3795 | e-ISSN 1873-1856)
Recommended Citation
Fickus, M., Jasper, J., King, E. J., & Mixon, D. G. (2018). Equiangular tight frames that contain regular simplices. Linear Algebra and Its Applications, 555, 98–138. https://doi.org/10.1016/j.laa.2018.06.004 arXiv.org version furnished at AFIT Scholar:
arXiv:1711.07081 [math.FA], https://doi.org/10.48550/arXiv.1711.07081
Comments
© Elsevier B.V. 2018
The final published version of record is hosted at Elsevier as a subscription-access article, and is accessible through the DOI link on this page.
This posting on AFIT Scholar provides the author manuscript version (submitted version), as posted at arXiv:1711.07081 [math.FA].
Date of arXiv submission: 19 Nov 2017
Shared in accordance with permissions listed at the Open Policy Finder for this journal.
Reviewed at MR3834195