Algorithms and error bounds for noisy phase retrieval with low-redundancy frames

TL;DR

This paper develops algorithms with explicit error bounds for noisy phase retrieval using low-redundancy frames, demonstrating polynomial-time methods effective under small noise conditions.

Contribution

It introduces explicit error bounds and efficient algorithms for phase retrieval with low-redundancy frames, a setting less explored in prior work.

Findings

Algorithms achieve polynomial time complexity.

Error bounds inversely proportional to signal-to-noise ratio.

Effective recovery under small noise conditions.

Abstract

The main objective of this paper is to find algorithms accompanied by explicit error bounds for phase retrieval from noisy magnitudes of frame coefficients when the underlying frame has a low redundancy. We achieve these goals with frames consisting of $N=6d-3$ vectors spanning a $d$-dimensional complex Hilbert space. The two algorithms we use, phase propagation or the kernel method, are polynomial time in the dimension $d$. To ensure a successful approximate recovery, we assume that the noise is sufficiently small compared to the squared norm of the vector to be recovered. In this regime, the error bound is inverse proportional to the signal-to-noise ratio. Upper and lower bounds on the sample values of trigonometric polynomials are a central technique in our error estimates.