Steiner equiangular tight frames

By Matthew C. Fickus, Dustin G. Mixon, Janet C. Tremain. 2010

214 citations recorded by the source corpus.

Abstract We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of infinite families of such frames. It provides great freedom in terms of the frame’s size and redundancy. This method also explicitly constructs the frame vectors in their native domain, as opposed to implicitly defining them via their Gram matrix. Moreover, in this domain, the frame vectors are very sparse. The construction is extremely simple: a tensor-like combination of a Steiner system and a regular simplex. This simplicity permits us to resolve an open question regarding ETFs and the restricted isometry property (RIP): we show that the RIP behavior of some ETFs is unfortunately no better than their coherence indicates.

Subfields: Applied Mathematics, Geophysics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/68e6953666b06a942460ede8949b26f0ef632602

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Abhinav Kumar
2. Henry Cohn
3. Suhas N. Diggavi
4. Yifan Sun
5. Gregory T. Minton
6. Can Bülent Karakus
7. Wotao Yin
8. Salvatore Torquato

In Geophysics

1. Suhas N. Diggavi
2. Yifan Sun
3. Abhinav Kumar
4. Gregory T. Minton
5. Henry Cohn
6. Can Bülent Karakus
7. Wotao Yin

In Signal Processing

1. Suhas N. Diggavi
2. Yifan Sun
3. Abhinav Kumar
4. Gregory T. Minton
5. Henry Cohn
6. Can Bülent Karakus
7. Wotao Yin
8. Jakub Konecný

Citations in the corpus

• Optimal simplices and codes in projective spaces (2013, methodology): “The Steiner construction can be carried out in CP 1 using a complex Hadamard matrix instead of a real Hadamard matrix (see [36]).”
• Redundancy Techniques for Straggler Mitigation in Distributed Optimization and Learning (2018, methodology): “We illustrate the described technique through Steiner ETF, based on the construction proposed in Fickus et al. (2012), using (2, 2, v)-Steiner systems.”
• Straggler Mitigation in Distributed Optimization Through Data Encoding (2017, background): “We also discuss Steiner ETFs [8] in Appendix D, which enable efficient implementation.”