Before revision:  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:  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 to provably achieve the same accuracy as the best one-dimensional algorithms in less time.