  In a mixed generalized linear model, the objective is to learn multiple
signals from unlabeled observations: each sample comes from exactly one signal,
but it is not known which one. We consider the prototypical problem of
estimating two statistically independent signals in a mixed generalized linear
model with Gaussian covariates. Spectral methods are a popular class of
estimators which output the top two eigenvectors of a suitable data-dependent
matrix. However, despite the wide applicability, their design is still obtained
via heuristic considerations, and the number of samples $n$ needed to guarantee
recovery is super-linear in the signal dimension $d$. In this paper, we develop
exact asymptotics on spectral methods in the challenging proportional regime in
which $n, d$ grow large and their ratio converges to a finite constant. By
doing so, we are able to optimize the design of the spectral method, and
combine it with a simple linear estimator, in order to minimize the estimation
error. Our characterization exploits a mix of tools from random matrices, free
probability and the theory of approximate message passing algorithms. Numerical
simulations for mixed linear regression and phase retrieval demonstrate the
advantage enabled by our analysis over existing designs of spectral methods.
