论文标题
通过phaseliftoff稀疏的相位检索
Sparse phase retrieval via Phaseliftoff
论文作者
论文摘要
稀疏期检索的目的是恢复$ K $ -SPARSE信号$ \ MATHBF {X} _0 \ in \ MathBb {C}^{d} $从Quadratic测量中$ \ mathbf {a} _i \ in \ mathbb {c}^d，i = 1，\ ldots，m $。注意$ | \ langle \ mathbf {a} _i，\ mathbf {x} _0 \ rangle |^2 = {\ text {trext {tr}}（a_ix_0）$ at $ a_i = \ \ \ \ \ \ \ m mathbf {a} a} \mathbb{C}^{d\times d}, X_0=\mathbf{x}_0\mathbf{x}_0^*\in \mathbb{C}^{d\times d}$, one can recast sparse phase retrieval as a problem of recovering a rank-one sparse matrix from linear measurements. Yin和Xin引入了Phaseliftoff，该法线通过痕量和Frobenius Norm的差异提出了级别状态的代理。通过向Phaseliftoff添加稀疏性惩罚，我们提出了一个新型模型，以从二次测量中恢复稀疏信号。理论分析表明，在几乎最佳采样复杂度下，我们模型的解决方案提供了$ \ mathbf {x} _0 $的稳定恢复，$ m = o（k \ log（d/k））$。我们的模型的计算是通过凸函数算法（DCA）的差进行的。数值实验表明，我们的算法优于用于解决稀疏相检索的其他最先进算法。
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.
