We study a spectral initialization method that serves a key role in recent\nwork on estimating signals in nonconvex settings. Previous analysis of this\nmethod focuses on the phase retrieval problem and provides only performance\nbounds. In this paper, we consider arbitrary generalized linear sensing models\nand present a precise asymptotic characterization of the performance of the\nmethod in the high-dimensional limit. Our analysis also reveals a phase\ntransition phenomenon that depends on the ratio between the number of samples\nand the signal dimension. When the ratio is below a minimum threshold, the\nestimates given by the spectral method are no better than random guesses drawn\nfrom a uniform distribution on the hypersphere, thus carrying no information;\nabove a maximum threshold, the estimates become increasingly aligned with the\ntarget signal. The computational complexity of the method, as measured by the\nspectral gap, is also markedly different in the two phases. Worked examples and\nnumerical results are provided to illustrate and verify the analytical\npredictions. In particular, simulations show that our asymptotic formulas\nprovide accurate predictions for the actual performance of the spectral method\neven at moderate signal dimensions.\n