Approximate Message Passing for Amplitude Based Optimization
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We consider an ℓ_2-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting m,n →∞, m/n →δ and obtain sharp performance bounds, where m is the number of measurements and n is the signal dimension. We show that for complex signals the algorithm can perform accurate recovery with only m= ( 64/π^2-4)n≈ 2.5n measurements. Also, we provide sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: (i) Adding ℓ_2 regularization to the non-convex loss function can be beneficial even in the noiseless setting; (ii) spectral initialization has marginal impact on the performance of the algorithm.
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