Compressive Phase Retrieval of Structured Signals

Abstract

Compressive phase retrieval is the problem of recovering a structured vector $x\in \mathbb{C}^{n}$ from its phaseless linear measurements. A compression algorithm aims to represent structured signals with as few bits as possible. As a result of extensive research devoted to compression algorithms, in many signal classes, compression algorithms are capable of employing sophisticated structures in signals and compress them efficiently. This raises the following important question: Can a compression algorithm be used to solve a compressive phase retrieval problem? To address this question, COmpressive PhasE Retrieval (COPER) optimization is proposed, which is a compression-based phase retrieval method. For a family of compression codes with rate-distortion function denoted by $r(\delta)$ , in the noiseless setting, COPER is shown to require slightly more than $\lim\limits_{\delta\rightarrow 0}\frac{r(\delta)}{\log(1/\delta)}$ observations for an almost accurate recovery of $x$ .