We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. Our work leverages recent results that provide an exact characterization of the performance of the spectral method in the highdimensional limit. This characterization allows us to map the task of optimal design to a constrained optimization problem in a weighted L 2 function space. The latter has a closed-form solution.Interestingly, under a mild technical condition, our results show that there exists a fixed design that is uniformly optimal over all sampling ratios. Numerical simulations demonstrate the performance improvement brought by the proposed optimal design over existing constructions in the literature. In a recent work, Mondelli and Montanari have shown the existence of a weak reconstruction threshold below which the spectral method cannot provide useful estimates. Our results serve to complement that work by deriving the fundamental limit of the spectral method beyond the aforementioned threshold.

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Construction of Capacity Achieving Lattice Gaussian Codes Wael AlghamdiWe propose a new approach to proving results regarding channel coding schemes based on construction−A lattices for the Additive White Gaussian Noise (AWGN) channel that yields new characterizations of the code construction parameters, i.e., the primes and dimensions of the codes, as functions of the block-length. The approach we take introduces an averaging argument that explicitly involves the considered parameters.

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