SUPER: Sparse signals with Unknown Phases Efficiently Recovered
(1401.4269)Abstract
Suppose ${\bf x}$ is any exactly $k$-sparse vector in $\mathbb{C}{n}$. We present a class of phase measurement matrix $A$ in $\mathbb{C}{m\times n}$, and a corresponding algorithm, called SUPER, that can resolve ${\bf x}$ up to a global phase from intensity measurements $|A{\bf x}|$ with high probability over $A$. Here $|A{\bf x}|$ is a vector of component-wise magnitudes of $A{\bf x}$. The SUPER algorithm is the first to simultaneously have the following properties: (a) it requires only ${\cal O}(k)$ (order-optimal) measurements, (b) the computational complexity of decoding is ${\cal O}(k\log k)$ (near order-optimal) arithmetic operations.
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