  We present the optimal design of a spectral method widely used to initialize
nonconvex optimization algorithms for solving phase retrieval and other signal
recovery problems. Our work leverages recent results that provide an exact
characterization of the performance of the spectral method in the
high-dimensional limit. This characterization allows us to map the task of
optimal design to a constrained optimization problem in a weighted $L^2$
function space. The latter has a closed-form solution. Interestingly, under a
mild technical condition, our results show that there exists a fixed design
that is uniformly optimal over all sampling ratios. Numerical simulations
demonstrate the performance improvement brought by the proposed optimal design
over existing constructions in the literature. In a recent work, Mondelli and
Montanari have shown the existence of a weak reconstruction threshold below
which the spectral method cannot provide useful estimates. Our results serve to
complement that work by deriving the fundamental limit of the spectral method
beyond the aforementioned threshold.
