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We tackle the problem of recovering a complex signal $\vx\in\mathbb{C}^n$ from quadratic measurements of the form $y_i=\vx^*\vA_i\vx$, where $\{\vA_i\}_{i=1}^m$ is a set of complex iid standard Gaussian matrices. This non-convex problem is related to the well understood phase retrieval problem where $\vA_i$ is a rank-1 positive semidefinite matrix. Here we study a general full-rank case which models a number of key applications such as molecular geometry recovery from distance distributions and compound measurements in phaseless diffractive imaging. Most prior work either addresses the rank-1 case or focuses on real measurements. The several papers that address the full-rank complex case adopt the semidefinite relaxation approach and are thus computationally demanding. In this paper we propose a method based on the standard framework comprising a spectral initialization followed by iterative gradient descent updates. We prove that when the number of measurements exceeds the signal's length by some constant factor, a globally optimal solution can be recovered from complex quadratic measurements with high probability. Numerical experiments on simulated data corroborate our theoretical analysis. 

In this paper, we study a 2D tomography problem for point source models with random unknown view angles. Rather than recovering the projection angles, we reconstruct the model through a set of rotation-invariant features that are estimated from the projection data. For a point source model, we show that these features reveal geometric information about the model such as the radial and pairwise distances. This establishes a connection between unknown view tomography and unassigned distance geometry problem (uDGP). We propose new methods to extract the distances and approximate the pairwise distance distribution of the underlying points. We then use the recovered distribution to estimate the locations of the points through constrained non-convex optimization. Our simulation results show that our point source reconstruction pipeline is robust to noise and outperforms the regularized expectation maximization (EM) baseline.