IMISCOE Research Series

Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal x∈CN in Poly(N)log⁡(1/ϵ) time to reach ϵ-precision from its magnitude of discrete Fourier transform and its initial value x(0), then P=NP. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value |x(0)|, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.

In this paper, we study the high probability convergence of AdaGrad-Norm for constrained, non-smooth, weakly convex optimization with bounded noise and sub-Gaussian noise cases. We also investigate a more general accelerated gradient descent (AGD) template (Ghadimi and Lan, 2016) encompassing the AdaGrad-Norm, the Nesterov's accelerated gradient descent, and the RSAG (Ghadimi and Lan, 2016) with different parameter choices. We provide a high probability convergence rate O˜(1/T) without knowing the information of the weak convexity parameter and the gradient bound to tune the step-sizes.