Almost Optimal Phaseless Compressed Sensing with Sublinear Decoding Time

Vasileios Nakos

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arXiv · 1701.06437

Almost Optimal Phaseless Compressed Sensing with Sublinear Decoding Time

Vasileios Nakos

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Abstract

In the problem of compressive phase retrieval, one wants to recover an approximately $k$-sparse signal $x \in \mathbb{C}^n$, given the magnitudes of the entries of $\Phi x$, where $\Phi \in \mathbb{C}^{m \times n}$. This problem has received a fair amount of attention, with sublinear time algorithms appearing in \cite{cai2014super,pedarsani2014phasecode,yin2015fast}. In this paper we further investigate the direction of sublinear decoding for real signals by giving a recovery scheme under the $\ell_2 / \ell_2$ guarantee, with almost optimal, $\Oh(k \log n )$, number of measurements. Our result outperforms all previous sublinear-time algorithms in the case of real signals. Moreover, we give a very simple deterministic scheme that recovers all $k$-sparse vectors in $\Oh(k^3)$ time, using $4k-1$ measurements.

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Vasileios Nakos. 2017-01-23. Almost Optimal Phaseless Compressed Sensing with Sublinear Decoding Time. https://arxiv.org/abs/1701.06437

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