Fourier phase retrieval, which seeks to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we give a theoretical understanding of algorithms for Fourier phase retrieval. Particularly, we show if there exists an algorithm which could reconstruct an arbitrary signal ${\mathbf x}\in {\mathbb C}^N$ in $ \mbox{Poly}(N) \log(1/\epsilon)$ time to reach $\epsilon$-precision from its magnitude of discrete Fourier transform and its initial value $x(0)$, then $\mathcal{ P}=\mathcal{NP}$. This demystifies the phenomenon that, although almost all signals are determined uniquely by their Fourier magnitude with a prior conditions, there is no algorithm with theoretical guarantees being proposed over the past few decades. Our proofs employ the result in computational complexity theory that Product Partition problem is NP-complete in the strong sense.

翻译：Fourier 阶段检索试图从 Fourier 级中重建信号,在工程和科学领域具有根本重要性。 在本文中,我们对Fourier 级检索的算法有了理论上的了解。 特别是, 我们显示是否有一种算法可以重建任意信号$_mathbf x ⁇ in {mathbb x ⁇ in {mathbb C ⁇ n$ $ $\ mbox{poly} (N) log(1/\\\ epsilon) $, 以便从离散的Fourier 变换规模和初始值$x( 0)$, 然后是$\mathcal{ P ⁇ mathcal{NP}$ 。 这个现象已经解析了, 尽管几乎所有信号都是由它们先前条件的四倍级所独有的, 但是在过去几十年里没有提出理论保证的算法。 我们的证据采用了计算复杂理论, 即产品分割部分问题在强烈意义上是完成 NPP。