No existence of linear algorithm for Fourier phase retrieval

TL;DR

This paper proves that no polynomial-time algorithm with guaranteed accuracy exists for Fourier phase retrieval unless P=NP, explaining the longstanding difficulty in developing such algorithms.

Contribution

It establishes a theoretical complexity barrier for Fourier phase retrieval, linking it to NP-completeness and showing the non-existence of efficient algorithms with guarantees.

Findings

Reconstructing signals from Fourier magnitude is NP-hard.

No polynomial-time algorithm can guarantee reconstruction unless P=NP.

The problem's complexity is connected to the NP-complete Product Partition problem.

Abstract

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.