Efficient Solutions for the Multidimensional Sparse Turnpike Problem
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The turnpike problem of recovering a set of points in ℝ^D from the set of their pairwise differences is a classic inverse problem in combinatorics. This problem or variants arise in a number of applications, prominently including bioinformatics and imaging, where it is intimately 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 ≥ 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(N^D(1/2 - δ)). Detailed and reproducible numerical experiments demonstrate the effectiveness of our algorithm, showing that in practice MISTR enjoys O(k^2logk) average-case time complexity, nearly linear in the size of the input. This represents a substantial improvement over existing one-dimensional algorithms.
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