Linear Spectral Estimators and an Application to Phase Retrieval
This work addresses phase retrieval, a problem in signal processing and imaging, by providing improved initialization for iterative algorithms, though it appears incremental as it builds on spectral initializers.
The authors tackled the problem of phase retrieval by proposing linear spectral estimators (LSPEs) to compute accurate initialization vectors, showing that these estimators significantly outperform existing methods for structured measurement systems in practice.
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.