稀疏多维相位恢复的支持度恢复
Support Recovery for Sparse Multidimensional Phase Retrieval
摘要：
我们考虑的\textit{phase retrieval}问题的恢复稀疏信号$\mathbf{x}$在$\mathbb{R}^d$从强度只有测量尺寸$d \geq 2$。相位恢复可以等效地表述为从其自相关恢复信号的问题，这又直接涉及从其成对差异恢复集合的组合问题。在一个空间维度中，该问题被充分研究并且被称为\textit{turnpike problem}。在这项工作中，我们提出了MISTR（多维交叉稀疏矩阵恢复），一种算法，利用这种配方来恢复支持的多维信号的幅度只有测量。MISTR利用多维结构的优势，可以证明在更短的时间内达到与最好的一维算法相同的精度。我们从理论上证明，MISTR正确地恢复支持分布为高斯点过程的信号具有很高的概率，只要稀疏度是最多$\mathcal{O}\left(n^{dθ}\right)$的任何$θ< 1/2$，其中$n^d$代表在一个固定的图像窗口中的像素大小。在幅度测量被噪声破坏的情况下，我们提供了一个阈值计划的理论保证稀疏最多$\mathcal{O}\left(n^{dθ}\right)$的$θ< 1/4$，避免了需要MISTR明确处理噪声自相关数据。详细的和可重复的数值实验表明，我们的算法的有效性，在实践中MISTR享有的时间复杂度是近线性的输入的大小。
英文摘要：
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.
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