A recently proposed convex formulation of the phase retrieval problem estimates the unknown signal by solving a simple linear program. This new scheme, known as PhaseMax, is computationally efficient compared to standard convex relaxation methods based on lifting techniques. In this paper, we present an exact performance analysis of PhaseMax under Gaussian measurements in the large system limit. In contrast to previously known performance bounds in the literature, our results are asymptotically exact and they also reveal a sharp phase transition phenomenon. Furthermore, the geometrical insights gained from our analysis led us to a novel nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm based on successive linearization and maximization over a polytope. This new algorithm, which we call PhaseLamp, has provably superior recovery performance over the original PhaseMax method.

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We consider a recently proposed convex formulation, known as the PhaseMax method, for solving the phase retrieval problem. Using the replica method from statistical mechanics, we analyze the performance of PhaseMax in the high-dimensional limit. Our analysis predicts the exact asymptotic performance of PhaseMax. In particular, we show that a sharp phase transition phenomenon takes place, with a simple analytical formula characterizing the phase transition boundary. This result shows that the oversampling ratio required by existing performance bounds in the literature can be significantly reduced. Numerical results confirm the validity of our replica analysis, showing that the theoretical predictions are in excellent agreement with the actual performance of the algorithm, even for moderate signal dimensions.

We study the problem of learning an unknown function using random feature models. Our main contribution is an exact asymptotic analysis of such learning problems with Gaussian data. Under mild regularity conditions for the feature matrix, we provide an exact characterization of the asymptotic training and generalization errors, valid in both the under-parameterized and over-parameterized regimes. The analysis presented in this paper holds for general families of feature matrices, activation functions, and convex loss functions. Numerical results validate our theoretical predictions, showing that our asymptotic findings are in excellent agreement with the actual performance of the considered learning problem, even in moderate dimensions. Moreover, they reveal an important role played by the regularization, the loss function and the activation function in the mitigation of the "double descent phenomenon" in learning.

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