Almost Optimal Phaseless Compressed Sensing with Sublinear Decoding Time
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 sublineartime 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 $4k1$ measurements.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.06437
 Bibcode:
 2017arXiv170106437N
 Keywords:

 Computer Science  Information Theory
 EPrint:
 The running time of the algorithm in the Appendix was made k^2 instead of k^3, and the number of rows was corrected to 6k2 from 4k2