The turnpike problem of recovering a set of points in \mathbb{R in 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 phase retrieval problem of recovering a signal from magnitude-only measurements of its Fourier transform. We propose the Multidimensional Intersection Sparse Phase Retrieval (MISTR)algorithm for solving the turnpike problem in dimension D \geq 2. By taking advantage of the structure of multiple dimensions , we are able to achieve the same accuracy as the best one-dimensional algorithms in  less time. We prove theoretically that MISTR correctly recovers most signals distributed as a Gaussian point process  as long as sparsity is at most O \left( N ^{ D(}\right)  1/2 - \delta)}\left( \right) . Detailed and reproducible numerical experiments demonstrate the effectiveness of our algorithm, showing that in practice MISTR enjoys O%DIFDELCMD < \left(%%% k^2\log{k}%DIFDELCMD < \right) %%% average-case time complexity , nearly linear in the size of the input . This represents a substantial improvement over existing one-dimensional algorithms .
We consider thephase retrieval problem of recovering a sparse signal \mathbf{x in 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 theturnpike 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 ^{ d\theta}\right) for any \theta < 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^{d\theta \right) for \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 %DIFDELCMD < \left(%%% }%DIFDELCMD < \right) %%% time complexity which is nearly linear in the size of the input .
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