Abstract
This paper establishes the restricted isometry property for a Gabor system generated by n 2 time–frequency shifts of a random window function in n dimensions. The sth order restricted isometry constant of the associated n × n 2 Gabor synthesis matrix is small provided that s ≤ c n 2/3 / log2 n. This bound provides a qualitative improvement over previous estimates, which achieve only quadratic scaling of the sparsity s with respect to n. The proof depends on an estimate for the expected supremum of a second-order chaos.
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Dedicated to Hans G. Feichtinger on occassion of his 60th birthday.
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Pfander, G.E., Rauhut, H. & Tropp, J.A. The restricted isometry property for time–frequency structured random matrices. Probab. Theory Relat. Fields 156, 707–737 (2013). https://doi.org/10.1007/s00440-012-0441-4
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DOI: https://doi.org/10.1007/s00440-012-0441-4
Keywords
- Compressed sensing
- Restricted isometry property
- Gabor system
- Time–frequency analysis
- Random matrix
- Chaos process