New Conditions for Sparse Phase Retrieval

Abstract

We consider the problem of sparse phase retrieval, where a $k$-sparse signal $\mathbf{x}\in {\mathbb{R}}^n\text{ (or }{\mathbb{C}}^n\text{)}$ is measured as $\mathbf{y}=|\mathbf{Ax}|,$ where $\mathbf{A}\in {\mathbb{R}}^{m\times n}\text{ (or }{\mathbb{C}}^{m\times n}\text{ respectively)}$ is a measurement matrix and $|\cdot |$ is the element-wise absolute value. For a real signal and a real measurement matrix $\mathbf{A}$, we show that $m=2k$ measurements are necessary and sufficient to recover $\mathbf{x}$ uniquely. For complex signal $\mathbf{x}\in {\mathbb{C}}^n$ and $\mathbf{A}\in {\mathbb{C}}^{m\times n}$, we show that $m=4k-2$ phaseless measurements are sufficient to recover $\mathbf{x}$. It is known that the multiplying constant $4$ in $m=4k-2$ cannot be improved.

6 pages

References in corpus (1)

• An algebraic characterization of injectivity in phase retrieval

Cited by in corpus (2)

• Homomorphic Sensing of Subspace Arrangements
• Fast Compressive Phase Retrieval from Fourier Measurements