Affine phase retrieval for sparse signals via $\ell_1$ minimization

TL;DR

This paper demonstrates that $\,\ell_1$ minimization effectively recovers sparse signals from affine phase measurements, requiring a logarithmic number of Gaussian measurements and providing stability under noise.

Contribution

It introduces a novel $\,\ell_1$ minimization approach for affine phase retrieval of sparse signals, establishing measurement bounds and stability results.

Findings

Recovery of all $k$-sparse signals with $O(k\log(en/k))$ measurements.

Stable reconstruction in noisy measurement scenarios.

Theoretical guarantees for real and complex signals.

Abstract

Affine phase retrieval is the problem of recovering signals from the magnitude-only measurements with a priori information. In this paper, we use the $\ell_1$ minimization to exploit the sparsity of signals for affine phase retrieval, showing that $O(k\log(en/k))$ Gaussian random measurements are sufficient to recover all $k$-sparse signals by solving a natural $\ell_1$ minimization program, where $n$ is the dimension of signals. For the case where measurements are corrupted by noises, the reconstruction error bounds are given for both real-valued and complex-valued signals. Our results demonstrate that the natural $\ell_1$ minimization program for affine phase retrieval is stable.