On the Existence of Equiangular Tight Frames | Center for Big Data Analytics

Abstract: An equiangular tight frame (ETF) is a d × N matrix that has unit-norm columns and orthogonal rows of norm sqrt(N/d). Its key property is that the absolute inner products between pairs of columns are (i) identical and (ii) as small as possible. ETFs have applications in communications, coding theory, and sparse approximation. Numerical experiments indicate that ETFs arise for very few pairs (d,N), and it is an important challenge to develop restrictions on the pairs for which they can exist. This article uses field theory to provide detailed conditions on real and complex ETFs. In particular, it describes restrictions on harmonic ETFs, a specific type of complex ETF that appears in applications. Finally, the article offers empirical evidence that these conditions are sharp or nearly sharp, especially in the real case.

Topics:
Tight Frames

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Citation

On the Existence of Equiangular Tight Frames (pdf, software)
M. Sustik, J. Tropp, I. Dhillon, R. Jr..
Linear Algebra and its Applications 426(2), pp. 619-635, October 2007.
(A previous version appears as UTCS Technical Report #TR-04-32, August 2004.)

Bibtex:
@article{sustik2007ontheexi, author = "Mátyás Sustik AND Joel A. Tropp AND Inderjit S. Dhillon AND Robert W. Heath Jr.", title = "On the Existence of Equiangular Tight Frames", journal = "Linear Algebra and its Applications", page = "619–635", volume = "426", issue = "2", number = "0", year = "2007", month = "oct", note = "; A previous version appears as UTCS Technical Report #TR-04-32, August 2004.", abstract = "An equiangular tight frame (ETF) is a d × N matrix that has unit-norm columns and orthogonal rows of norm sqrt(N/d). Its key property is that the absolute inner products between pairs of columns are (i) identical and (ii) as small as possible. ETFs have applications in communications, coding theory, and sparse approximation. Numerical experiments indicate that ETFs arise for very few pairs (d,N), and it is an important challenge to develop restrictions on the pairs for which they can exist. This article uses field theory to provide detailed conditions on real and complex ETFs. In particular, it describes restrictions on harmonic ETFs, a specific type of complex ETF that appears in applications. Finally, the article offers empirical evidence that these conditions are sharp or nearly sharp, especially in the real case." }