Sparse Phase Retrieval Via PhaseLiftOff

The aim of sparse phase retrieval is to recover a k-sparse signal x(0) is an element of C-d from quadratic measurements vertical bar < a(j), x(0)>vertical bar(2) where a(j) is an element of C-d, j = 1, ..., m. Noting vertical bar < a(j), x(0)>vertical bar(2) = Tr(A(j)X(0)) with A(j) = a(j)a(j)* is an element of C-dxd, X-0 = x(0)x(0)* is an element of C-dxd, 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 optimal solution to our model provides the stable recovery of x(0) under almost optimal sampling complexity m = O(k log(d/k)). We use the difference of convex function algorithm (DCA) to solve PhaseLiftOff, showing DCA converges to a stationary point. Numerical experiments demonstrate that our algorithm outperforms other state-of-the-art algorithms used for solving sparse phase retrieval.