SUPER: Sparse signals with Unknown Phases Efficiently Recovered

TL;DR

The paper introduces SUPER, an efficient algorithm and measurement matrix for recovering exactly sparse complex signals from intensity-only measurements, achieving near-optimal measurement count and computational complexity.

Contribution

It presents the first method that uses only O(k) measurements and O(k log k) decoding complexity for phase retrieval of sparse signals.

Findings

Requires only O(k) measurements, which is order-optimal.

Decoding complexity is O(k log k), near order-optimal.

Successfully recovers sparse signals with high probability.

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.