A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant ∈ if it preserves the l2 norm of all k-sparse vectors up to a factor of 1 ± ϵ. We prove that a matrix A obtained by randomly sampling q = O(k log2 k log N) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ∈ with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).
CITATION STYLE
Haviv, I., & Regev, O. (2017). The restricted isometry property of subsampled fourier matrices. In Lecture Notes in Mathematics (Vol. 2169, pp. 163–179). Springer Verlag. https://doi.org/10.1007/978-3-319-45282-1_11
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