Before revision: 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.
After revision: 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 set from its pairwise differences. In one spatial dimension, this problem is well studied and known as 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.