论文标题
基于振幅的模型用于复杂阶段检索的性能
The performance of the amplitude-based model for complex phase retrieval
论文作者
论文摘要
本文旨在研究基于振幅的模型的性能\ newline $ \ wideHat {\ MathBf x} \ in {\ rm argmin} _ {{\ rm argmin} _ {{\ mathbf x} \ in \ mathbb {c} a} _j，{\ mathbf x} \ rangle | -b_j \ right）^2 $，其中$ b_j：= | \ langle {\ mathbf a} _j，{\ mathbf x} _0 _0 _0 \ rangle |是目标信号。该模型在相位检索以及绝对值整流神经网络中提出。在过去的几十年中，已经开发了许多有效的算法来解决它。 {但是，在嘈杂条件下复杂情况下的估计性能的结果很少。}在本文中，{我们在基于振幅的复杂相位检索问题的基于幅度的模型上提供了理论保证。具体而言，我们表明$ \ min_ {θ\ in [0,2π）}} \ | \ | \ wideHat {\ MathBf x} - \ exp（\ mathrm {i}θ）\ cdot {\ cdot {\ cdot {\ mathbf x} _0 η} \ | _2} {\ sqrt {m}} $具有很高的概率，前提是测量向量$ {\ mathbf a} _j \ in \ in \ in \ mathbb {c}^d，$ $ $ j = 1，$ j = 1，\ ldots，\ ldots，\ ldots，m，m，$ ranty complect和$ i.i.imim {mimi.mmim {mimim {mimi} D $。这里$ {\mathbfη} =（η_1，\ ldots，η_m）\ in \ mathbb {r}^m $是噪声向量，而没有任何假设。此外，我们证明了重建误差是明显的。对于目标信号$ {\ mathbf x} _0 \ in \ mathbb {c}^{d} $很少的情况，我们为非线性约束$ \ ell_1 $最小化模型建立了相似的结果。 {为了实现这一目标，我们利用了在同时低级和稀疏矩阵的空间上为操作员的限制性等轴测属性的强大版本。}
The paper aims to study the performance of the amplitude-based model \newline $\widehat{\mathbf x} \in {\rm argmin}_{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$, where $b_j:=|\langle {\mathbf a}_j,{\mathbf x}_0\rangle|+η_j$ and ${\mathbf x}_0\in \mathbb{C}^d$ is a target signal. The model is raised in phase retrieval as well as in absolute value rectification neural networks. Many efficient algorithms have been developed to solve it in the past decades. {However, there are very few results available regarding the estimation performance in the complex case under noisy conditions.} In this paper, {we present a theoretical guarantee on the amplitude-based model for the noisy complex phase retrieval problem}. Specifically, we show that $\min_{θ\in[0,2π)}\|\widehat{\mathbf x}-\exp(\mathrm{i}θ)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf η}\|_2}{\sqrt{m}}$ holds with high probability provided the measurement vectors ${\mathbf a}_j\in \mathbb{C}^d,$ $j=1,\ldots,m,$ are {i.i.d.} complex sub-Gaussian random vectors and $m\gtrsim d$. Here ${\mathbf η}=(η_1,\ldots,η_m)\in \mathbb{R}^m$ is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal ${\mathbf x}_0\in \mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear constrained $\ell_1$ minimization model. { To accomplish this, we leverage a strong version of restricted isometry property for an operator on the space of simultaneous low-rank and sparse matrices.}
