# Almost Optimal Phaseless Compressed Sensing with Sublinear Decoding Time

## TL;DR

This paper presents a nearly optimal sublinear-time algorithm for phase retrieval of real sparse signals with minimal measurements, improving upon previous methods and offering a simple deterministic recovery scheme.

## Contribution

It introduces a sublinear decoding scheme for real signals with near-optimal measurements and a simple deterministic method for exact sparse recovery.

## Key findings

- Achieves $ ext{O}(k ext{log} n)$ measurements for phase retrieval.
- Outperforms previous sublinear algorithms for real signals.
- Provides a deterministic recovery algorithm with $ ext{O}(k^3)$ time and $4k-1$ measurements.

## 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.