Abstract
Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term numerically erasure-robust frames was introduced in Fickus and Mixon (Linear Algebra Appl 437:1394–1407, 2012) to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15 % erasures, and asked whether there exist frames that are robust against erasures of more than 50 %. In this paper we show that with very high probability random frames are, independent of the dimension, robust against erasures as long as the number of remaining vectors is at least \(1+\delta _0\) times the dimension for some \(\delta _0>0\). This is the best possible result, and it also implies that the proportion of erasures can be arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.
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Acknowledgments
This research was completed while the author was at Department of Mathematics, Michigan State University. Part of this research was supported in part by the National Science Foundation grant DMS-1043032 and AFOSR grant FA9550-12-1-0455.
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Communicated by Hans G. Feichtinger.
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Wang, Y. Random Matrices and Erasure Robust Frames. J Fourier Anal Appl 24, 1–16 (2018). https://doi.org/10.1007/s00041-016-9486-6
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DOI: https://doi.org/10.1007/s00041-016-9486-6
Keywords
- Random matrices
- Singular values
- Numerically erasure robust frame (NERF)
- Condition number
- Restricted isometry property