Equiangular tight frames that contain regular simplices
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 th...
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| Published in | Linear algebra and its applications Vol. 555; pp. 98 - 138 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier Inc
15.10.2018
American Elsevier Company, Inc |
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| 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. |
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| AbstractList | 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. |
| Author | Jasper, John Fickus, Matthew King, Emily J. Mixon, Dustin G. |
| Author_xml | – sequence: 1 givenname: Matthew surname: Fickus fullname: Fickus, Matthew email: Matthew.Fickus@gmail.com organization: Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, United States of America – sequence: 2 givenname: John surname: Jasper fullname: Jasper, John organization: Department of Mathematics and Statistics, South Dakota State University, Brookings, SD 57007, United States of America – sequence: 3 givenname: Emily J. surname: King fullname: King, Emily J. organization: Department of Mathematics and Computer Science, University of Bremen, Bremen, 28359, Germany – sequence: 4 givenname: Dustin G. surname: Mixon fullname: Mixon, Dustin G. organization: Department of Mathematics, Ohio State University, Columbus, OH 43210, United States of America |
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| Snippet | 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... |
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| SubjectTerms | Algorithms Equiangular Euclidean geometry Euclidean space Frame Integer programming Linear algebra Packing problem Subspaces Tight |
| Title | Equiangular tight frames that contain regular simplices |
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