Solving Complex Quadratic Systems with Full-Rank Random Matrices
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We tackle the problem of recovering a complex signal x∈C^n from quadratic measurements of the form y_i= x^* A_i x, where { A_i}_i=1^m is a set of full-rank, complex random matrices with rotation invariant entries. This non-convex problem is related to the well understood phase retrieval problem where A_i is a rank-1 positive semidefinite matrix. Here we study the 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 tackles specific measurement models and either addresses the rank-1 case or focuses on real measurements. The several papers that address the full-rank complex case adopt the computationally-demanding semidefinite relaxation approach. In this paper we prove that the general class of rotation invariant measurement models can be efficiently solved with high probability via the standard framework comprising a spectral initialization followed by iterative gradient descent updates. Numerical experiments on simulated data generated from the rotation invariant exponential family corroborate our theoretical analysis and provide insights in designing suitable measurement matrices.
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