Sparse phase retrieval via Phaseliftoff

08/20/2020

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by   Yu Xia, et al.

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The aim of sparse phase retrieval is to recover a k-sparse signal 𝐱_0∈ℂ^d from quadratic measurements |⟨𝐚_i,𝐱_0⟩|^2 where 𝐚_i∈ℂ^d, i=1,…,m. Noting |⟨𝐚_i,𝐱_0⟩|^2=Tr(A_iX_0) with A_i=𝐚_i𝐚_i^*∈ℂ^d× d, X_0=𝐱_0𝐱_0^*∈ℂ^d× 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 𝐱_0 under almost optimal sampling complexity m=O(klog(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.