We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly corrupted by arbitrary outliers. This problem arises from applications, such as phase retrieval, covariance sketching, quantum space tomography, and power spectrum estimation. We first propose a convex optimization algorithm that seeks the PSD matrix with the minimum @math -norm of the observation residual. The advantage of our algorithm is that it is free of parameters, therefore, eliminating the need for tuning parameters and allowing easy implementations. We establish that with high probability, a low-rank PSD matrix can be exactly recovered as soon as the number of measurements is large enough, even when a fraction of the measurements are corrupted by outliers with arbitrary magnitudes. Moreover, the recovery is also stable against bounded noise. With the additional information of an upper bound of the rank of the PSD matrix, we propose another nonconvex algorithm based on subgradient descent that demonstrates excellent empirical performance in terms of computational efficiency and accuracy.
In the absence of outliers, the PhaseLift algorithm in the following form where @math denotes the trace of @math , has been proposed to solve the phase retrieval problem @cite_13 @cite_25 @cite_0 . Later the same algorithm has been employed to recover low-rank PSD matrices in @cite_30 , where an order of @math measurements obtained from i.i.d. sub-Gaussian sensing vectors are shown to guarantee exact recovery in the noise-free case and stable recovery with bounded noise. One problem with the algorithm is that the noise bound @math is assumed known. Furthermore, it is not amenable to handle outliers, since @math can be arbitrarily large with outliers and consequently the ground truth @math quickly becomes infeasible for .
@cite_0 @cite_30 @cite_13 @cite_25
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Recovering an unknown complex signal from the magnitude of linear combinations of the signal is referred to as phase retrieval. We present an exact performance analysis of a recently proposed convex-optimization-formulation for this problem, known as PhaseMax. Standard convex-relaxation-based methods in phase retrieval resort to the idea of “lifting” which makes them computationally inefficient, since the number of unknowns is effectively squared. In contrast, PhaseMax is a novel convex relaxation that does not increase the number of unknowns. Instead it relies on an initial estimate of the true signal which must be externally provided. In this paper, we investigate the required number of measurements for exact recovery of the signal in the large system limit and when the linear measurement matrix is random with iid standard normal entries. If @math denotes the dimension of the unknown complex signal and @math the number of phaseless measurements, then in the large system limit, @math measurements is necessary and sufficient to recover the signal with high probability, where @math is the angle between the initial estimate and the true signal. Our result indicates a sharp phase transition in the asymptotic regime which matches the empirical result in numerical simulations. The paper considers the phase retrieval problem in N-dimensional complex vector spaces. It provides two sets of deterministic measurement vectors which guarantee signal recovery for all signals, excluding only a specific subspace and a union of subspaces, respectively. A stable analytic reconstruction procedure of low complexity is given. Additionally it is proven that signal recovery from these measurements can be solved exactly via a semidefinite program. A practical implementation with 4 deterministic diffraction patterns is provided and some numerical experiments with noisy measurements complement the analytic approach. We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. In quantum information, the orbit is used extensively and is known as the set of stabilizer states. We argue that due to their rich geometric structure and their near-optimal recovery properties, stabilizer states form an ideal model for structured measurements for phase retrieval. Our results hold for @math measurements, where the oversampling factor k varies between @math and @math depending on the orbit. The reconstruction is stable towards both additive noise and deviations from the assumption of low rank. If the matrices of interest are in addition positive semidefinite, reconstruction may be performed by a simple constrained least squares regression. Our proof methods could be adapted to cover orbits of other groups. Phase retrieval seeks to recover a signal @math x ? C p from the amplitude @math | A x | of linear measurements @math A x ? C n . We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in Gerchberg and Saxton (Optik 35:237---246, 1972), where each iteration is a matrix vector product. Numerical results show the performance of this approach over three different phase retrieval problems, in comparison with greedy phase retrieval algorithms and matrix completion formulations.