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Phase retrieval refers to the problem of recovering real- or complex-valuedvectors from magnitude measurements. The best-known algorithms for this problemare iterative in nature and rely on so-called spectral initializers thatprovide accurate initialization vectors. We propose a novel class of estimatorssuitable for general nonlinear measurement systems, called linear spectralestimators (LSPEs), which can be used to compute accurate initializationvectors for phase retrieval problems. The proposed LSPEs not only provideaccurate initialization vectors for noisy phase retrieval systems withstructured or random measurement matrices, but also enable the derivation ofsharp and nonasymptotic mean-squared error bounds. We demonstrate the efficacyof LSPEs on synthetic and real-world phase retrieval problems, and show thatour estimators significantly outperform existing methods for structuredmeasurement systems that arise in practice.

Linear Spectral Estimators and an Application to Phase Retrieval