基于Phaseliftoff的稀疏相位恢复
Sparse phase retrieval via Phaseliftoff
摘要：
稀疏相位恢复的目的是从二次测量值$|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2$恢复$k$ -稀疏信号$\mathbf{x}_0\in \mathbb{C}^{d}$，其中$\mathbf{a}_i\in \mathbb{C}^d, i=1,\ldots,m$。注意到具有$A_i=\mathbf{a}_i\mathbf{a}_i^*\in \mathbb{C}^{d\times d}, X_0=\mathbf{x}_0\mathbf{x}_0^*\in \mathbb{C}^{d\times d}$的$|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2={\text{Tr}}(A_iX_0)$，可以将稀疏相位恢复重铸为从线性测量恢复秩一稀疏矩阵的问题。Yin和Xin通过迹和Frobenius范数的差异引入了PhaseLiftOff，它代表了秩1条件的代理。通过在PhaseLiftOff中加入稀疏惩罚，本文提出了一种新的模型来从二次测量中恢复稀疏信号。理论分析表明，我们的模型的解决方案提供了几乎最佳的采样复杂度$m=O(k\log(d/k))$下$\mathbf{x}_0$的稳定恢复。模型的计算采用差分凸函数算法（DCA）。数值实验表明，我们的算法优于其他国家的最先进的算法用于解决稀疏相位检索。
英文摘要：
The aim of sparse phase retrieval is to recover a $k$-sparse signal $\mathbf{x}_0\in \mathbb{C}^{d}$ from quadratic measurements $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2$ where $\mathbf{a}_i\in \mathbb{C}^d, i=1,\ldots,m$. Noting $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2={\text{Tr}}(A_iX_0)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
