不存在一维傅立叶相位检索的线性算法

IF 1.8 2区 数学 Q1 MATHEMATICS
Meng Huang , Zhiqiang Xu
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引用次数: 0

摘要

傅立叶相位检索旨在从傅立叶幅度重建信号,在工程和科学领域具有重要的基础性意义。在本文中,我们从理论上理解了一维傅立叶相位检索问题的算法。具体地说,我们证明了如果存在一种算法,能在 Poly(N)log(1/ϵ) 时间内从离散傅里叶变换的幅度及其初始值 x(0) 重构任意信号 x∈CN,达到ϵ精度,那么 P=NP。这部分解释了一个现象,即尽管几乎所有信号都是由其傅里叶幅值及其初始值 |x(0)| 的绝对值唯一决定的,但在过去几十年中,还没有人提出理论上可以保证的算法。我们的证明采用了计算复杂性理论的结果,即乘积分割问题在强意义上是 NP-完全的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
No existence of a linear algorithm for the one-dimensional Fourier phase retrieval

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 xCN 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.

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来源期刊
Journal of Complexity
Journal of Complexity 工程技术-计算机:理论方法
CiteScore
3.10
自引率
17.60%
发文量
57
审稿时长
>12 weeks
期刊介绍: The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.
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