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 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.

arXiv commentThe running time of the algorithm in the Appendix was made k^2 instead of k^3, and the number of rows was corrected to 6k-2 from 4k-2