Title: Linear Spectral Estimators and an Application to Phase Retrieval,
Abstract: Phase retrieval refers to the problem of recovering real- or complex-valued
vectors from magnitude measurements. The best-known algorithms for this problem
are iterative in nature and rely on so-called spectral initializers that
provide accurate initialization vectors. We propose a novel class of estimators
suitable for general nonlinear measurement systems, called linear spectral
estimators (LSPEs), which can be used to compute accurate initialization
vectors for phase retrieval problems. The proposed LSPEs not only provide
accurate initialization vectors for noisy phase retrieval systems with
structured or random measurement matrices, but also enable the derivation of
sharp and nonasymptotic mean-squared error bounds. We demonstrate the efficacy
of LSPEs on synthetic and real-world phase retrieval problems, and show that
our estimators significantly outperform existing methods for structured
measurement systems that arise in practice.