## Huiping Li, Yu Xia, Phase retrieval with sub-Gaussian measurements via Riemannian optimization

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DOI: 10.23952/jano.3.2021.3.02
Volume 3, Issue 3, 31 December 2021, Pages 457-478

Abstract. This paper concerns the phase retrieval problem under random sub-Gaussian measurements. We propose one type of gradient descent method based on Riemannian optimization, namely, truncated Riemannian gradient descent algorithm (TRGrad), to deal with the sub-gaussian phase retrieval problem. Compared with traditional methods, the careful selection rule in our work ensures a tighter initial guess. The sequence generated by the TRGrad converges to the true solution $x\in\mathbb{R}^{n}$ at a geometric rate with high probability provided that the number of measurements $m= O(n)$. This implies that the sample complexity is optimal. In addition, several numerical experiments are provided to show the effectiveness and stability of the TRGrad, and demonstrates that the TRGrad performs better than the state-of-the-art methods, such as Wirtinger Flow (WF) algorithm, and Generalized WF algorithm.