Equiangular Tight Frames That Contain Regular Simplices

Authors

Matthew C. Fickus, Air Force Institute of TechnologyFollow
John Jasper, South Dakota State University
Emily J. King, University of Bremen
Dustin G. Mixon, The Ohio State University

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.

Comments

© Elsevier B.V. 2018

The "Link to Full Text" on this page opens the full article as hosted at the publisher's website.

Articles published under an Elsevier user license are protected by copyright. Users may access, download, copy, translate, text and data mine (but may not redistribute, display or adapt) the articles for non-commercial purposes provided that users: 

• Cite the article using an appropriate bibliographic citation (i.e. author(s), journal, article title, volume, issue, page numbers, DOI and the link to the definitive published version on ScienceDirect) 
• Maintain the integrity of the article 
• Retain copyright notices and links to these terms and conditions so it is clear to other users what can and cannot be done with the article 
• Ensure that, for any content in the article that is identified as belonging to a third party, any re-use complies with the copyright policies of that third party 
• Any translations, for which a prior translation agreement with Elsevier has not been established, must prominently display the statement: "This is an unofficial translation of an article that appeared in an Elsevier publication. Elsevier has not endorsed this translation." 

 This record previously pointed at the arXiv.org preprint of the article at arXiv:1711.07081 [math.FA]

Reviewed at MR3834195

DOI

10.1016/j.laa.2018.06.004

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