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The problem of recovering a signal $\mathbf{x}\in \mathbb{R}^n$ from a set ofmagnitude measurements $y_i=|\langle \mathbf{a}_i, \mathbf{x} \rangle |, \;i=1,\ldots,m$ is referred as phase retrieval, which has many applications infields of physical sciences and engineering. In this paper we show that thesmoothed amplitude flow model for phase retrieval has benign geometricstructure under the optimal sampling complexity. In particular, we show thatwhen the measurements $\mathbf{a}_i\in \mathbb{R}^n$ are Gaussian randomvectors and the number of measurements $m\ge Cn$, our smoothed amplitude flowmodel has no spurious local minimizers with high probability, ie., the targetsolution $\mathbf{x}$ is the unique global minimizer (up to a global phase) andthe loss function has a negative directional curvature around each saddlepoint. Due to this benign geometric landscape, the phase retrieval problem canbe solved by the gradient descent algorithms without spectral initialization.Numerical experiments show that the gradient descent algorithm with randominitialization performs well even comparing with state-of-the-art algorithmswith spectral initialization in empirical success rate and convergence speed.

Solving phase retrieval with random initial guess is nearly as good as by spectral initialization