We consider the sparse phase retrieval problem of recovering a sparse signal x\mathbf {x} in Rd\mathbb {R}^d from intensity-only measurements in dimension d≥2d \geq 2. Sparse phase retrieval can often 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 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 O(ndθ)\mathcal {O}(n^{d\theta }) for any θ<1/2\theta < 1/2, where ndn^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 O(ndθ)\mathcal {O}(n^{d\theta }) for θ<1/4\theta < 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.

中文翻译：

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支持稀疏多维相位检索的恢复


我们考虑从维度 d≥2d \geq 2 中的仅强度测量中恢复 Rd\mathbb {R}^d 中的稀疏信号 x\mathbf {x} 的稀疏相位检索问题。稀疏相位检索通常可以等效地表述为从自相关中恢复信号的问题，这又与从成对差异中恢复集合的组合问题直接相关。在一个空间维度上，这个问题得到了很好的研究，被称为收费公路问题。在这项工作中，我们提出了 MISTR（多维交集稀疏支持恢复），这是一种利用该公式从仅幅度测量中恢复多维信号支持的算法。 MISTR 利用多维结构，可在极短的时间内达到与最佳一维算法相同的精度。我们从理论上证明，只要对于任何 θ<1 稀疏度至多为 O(ndθ)\mathcal {O}(n^{d\theta })，MISTR 就能以高概率正确恢复作为高斯点过程分布的信号的支持/2\theta < 1/2，其中 ndn^d 表示固定图像窗口中的像素大小。在幅度测量被噪声破坏的情况下，我们提供了一种阈值方案，理论上保证稀疏度最多为 O(ndθ)\mathcal {O}(n^{d\theta }) for θ<1/4\theta % 3C 1/4 消除了 MISTR 显式处理噪声自相关数据的需要。详细且可重复的数值实验证明了我们算法的有效性，表明实际上 MISTR 的时间复杂度与输入的大小几乎呈线性关系。