| Abstract: |
| We investigate the problem of generalized phase retrieval, which entails reconstructing a signal from its phaseless samples. We demonstrate that several well-known phase retrieval algorithms actually solve the least-squares fitting of lifted linear equations within the manifold of rank-1 matrices, using different Riemannian metrics on the manifold. However, these metrics only permit a stable and distinctly non-isometric embedding of rank-1 matrices, leading to linear convergence with a notably slow rate. To accelerate the convergence, we introduce a new metric on the rank-1 matrix manifold that enables an almost isometric embedding of these matrices. We propose a Riemannian gradient descent (RGD) algorithm, named Weighted RGD (WRGD), with this new metric to solve the phase retrieval problem. Due to the near isometric nature of this metric, we demonstrate that our WRGD, initialized via the spectral method, achieves linear convergence to the target signal at an enhanced rate. |
|