The aim of sparse phase retrieval is to recover a k-sparse signal x0 ∈ Cd from quadratic measurements |(aj,x0)|2 where aj ∈ ℂd, j=1,⋯,m. Noting |(aj,x0)|2=Tr(AjX0) with Aj=ajaj* ∈ ℂd×d, X0=x0x0* ∈ ℂd×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 optimal solution to our model provides the stable recovery of x0 under almost optimal sampling complexity m=O(klog(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.

中文翻译：

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通过 PhaseLiftOff 进行稀疏相位检索


稀疏相位检索的目的是从二次测量 |(aj,x0)|2 恢复 k 稀疏信号 x0 ∈ Cd，其中 aj ∈ ℂd, j=1,⋯,m。注意到 |(aj,x0)|2=Tr(AjX0) 且 Aj=ajaj* ε ℂd×d, X0=x0x0* ε ℂd×d，我们可以将稀疏相位检索重新定义为恢复一阶稀疏矩阵的问题来自线性测量。 Yin和Xin介绍了PhaseLiftOff，它通过迹线和Frobenius范数的差异来表示一级条件的代理。在本文中，通过向 PhaseLiftOff 添加稀疏性惩罚，我们提出了一种从二次测量中恢复稀疏信号的新颖模型。理论分析表明，我们模型的最优解在几乎最优的采样复杂度 m=O(klog(d/k)) 下提供了 x0 的稳定恢复。我们使用凸函数差分算法（DCA）来求解 PhaseLiftOff，表明 DCA 收敛到驻点。数值实验表明，我们的算法优于用于解决稀疏相位检索的其他最先进算法。