Improved Lower Bounds for the Restricted Isometry Property of Subsampled Fourier Matrices

Shravas Rao

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arXiv · 1903.12146

Improved Lower Bounds for the Restricted Isometry Property of Subsampled Fourier Matrices

Shravas Rao

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Abstract

Let $A$ be an $N \times N$ Fourier matrix over $\mathbb{F}_p^{\log{N}/\log{p}}$ for some prime $p$. We improve upon known lower bounds for the number of rows of $A$ that must be sampled so that the resulting matrix $M$ satisfies the restricted isometry property for $k$-sparse vectors. This property states that $\|Mv\|_2^2$ is approximately $\|v\|_2^2$ for all $k$-sparse vectors $v$. In particular, if $k = Ω( \log^2{N})$, we show that $Ω(k\log{k}\log{N}/\log{p})$ rows must be sampled to satisfy the restricted isometry property with constant probability.

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Shravas Rao. 2019-03-28. Improved Lower Bounds for the Restricted Isometry Property of Subsampled Fourier Matrices. https://arxiv.org/abs/1903.12146

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