New Conditions for Sparse Phase Retrieval

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