Sparse phase retrieval via Phaseliftoff

Abstract: The aim of sparse phase retrieval is to recover a $k$-sparse signal $\mathbf{x}_0\in \mathbb{C}<sup>{d}$ from quadratic measurements $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|<sup>2$ where $$\mathbf{a}_i\in \mathbb{C}^d, i=1,\ldots,m$$. Noting $$|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|<sup>2={\text{Tr}}(A_iX_0)$$ with $$A_i=\mathbf{a}_i\mathbf{a}_i^*\in \mathbb{C}^{d\times d}, X_0=\mathbf{x}_0\mathbf{x}_0^*\in \mathbb{C}^{d\times d}$$, one can recast sparse phase retrieval as a problem of recovering a rank-one sparse matrix from linear measurements. Yin and Xin introduced PhaseLiftOff which presents a proxy of rank-one condition via the difference of trace and Frobenius norm. By adding sparsity penalty to PhaseLiftOff, in this paper, we present a novel model to recover sparse signals from quadratic measurements. Theoretical analysis shows that the solution to our model provides the stable recovery of $\mathbf{x}_0$ under almost optimal sampling complexity $m=O(k\log(d/k))$. The computation of our model is carried out by the difference of convex function algorithm (DCA). Numerical experiments demonstrate that our algorithm outperforms other state-of-the-art algorithms used for solving sparse phase retrieval.