论文标题
支持稀疏多维阶段检索的恢复
Support Recovery for Sparse Multidimensional Phase Retrieval
论文作者
论文摘要
我们考虑\ textit {phase reterieval}在$ \ mathbb {r}^d $中恢复稀疏信号$ \ mathbf {x} $从dimension $ d \ geq 2 $中的仅限测量中的问题。相位检索可以等效地表达为从其自相关中恢复信号的问题，这反过来又与从其成对差异中恢复集合的组合问题直接相关。在一个空间维度中，对此问题进行了充分的研究，并称为\ textit {收费公路问题}。在这项工作中，我们介绍了MISTR（多维交叉点稀疏支持恢复），该算法利用了该公式以从仅尺度测量值中恢复多维信号的支持。 MISTR利用多个维度的结构，可以证明与最佳的一维算法相同的精度，在较小的时间上。从理论上讲，如果Mistr正确地恢复了分布在高斯点过程的信号的支持下，只要稀疏性最多是$ \ Mathcal {o} \ left（n^{dθ}} \ right（n^{dθ} \ right）$，对于任何$θ<1/2 $，其中$ n^d $代表$ n^d $代表pixents pixels in n picte n ofice of in ote n pixent pixel siper necte tecent of in ote n picixel。如果幅度测量值被噪声损坏，我们提供了一个阈值方案，并具有理论上保证了稀疏性的$ \ Mathcal {o} \ left（n^{dθ}} \ right）$ for $θ<1/4 $，从而证明了对误解的误解需要误解了噪音噪声的自动增压数据。详细且可重现的数值实验证明了我们的算法的有效性，表明实际上MISTR具有时间复杂性，这几乎是在输入大小的线性。
We consider the \textit{phase retrieval} problem of recovering a sparse signal $\mathbf{x}$ in $\mathbb{R}^d$ from intensity-only measurements in dimension $d \geq 2$. Phase retrieval can be equivalently formulated as the problem of recovering a signal from its autocorrelation, which is in turn directly related to the combinatorial problem of recovering a set from its pairwise differences. In one spatial dimension, this problem is well studied and known as the \textit{turnpike problem}. In this work, we present MISTR (Multidimensional Intersection Sparse supporT Recovery), an algorithm which exploits this formulation to recover the support of a multidimensional signal from magnitude-only measurements. MISTR takes advantage of the structure of multiple dimensions to provably achieve the same accuracy as the best one-dimensional algorithms in dramatically less time. We prove theoretically that MISTR correctly recovers the support of signals distributed as a Gaussian point process with high probability as long as sparsity is at most $\mathcal{O}\left(n^{dθ}\right)$ for any $θ< 1/2$, where $n^d$ represents pixel size in a fixed image window. In the case that magnitude measurements are corrupted by noise, we provide a thresholding scheme with theoretical guarantees for sparsity at most $\mathcal{O}\left(n^{dθ}\right)$ for $θ< 1/4$ that obviates the need for MISTR to explicitly handle noisy autocorrelation data. Detailed and reproducible numerical experiments demonstrate the effectiveness of our algorithm, showing that in practice MISTR enjoys time complexity which is nearly linear in the size of the input.
