Support Recovery in the Phase Retrieval Model: Information-Theoretic Fundamental Limits
This work addresses a foundational problem in signal processing and imaging applications like X-ray crystallography, offering theoretical limits for sparse variable identification, but it is incremental as it builds on existing phase retrieval models with new bounds.
The paper tackles the support recovery problem in the phase retrieval model with noisy phaseless measurements, deriving information-theoretic fundamental limits and providing sharp thresholds with near-matching constant factors for sparsity and signal-to-noise ratio scaling regimes.
The support recovery problem consists of determining a sparse subset of variables that is relevant in generating a set of observations. In this paper, we study the support recovery problem in the phase retrieval model consisting of noisy phaseless measurements, which arises in a diverse range of settings such as optical detection, X-ray crystallography, electron microscopy, and coherent diffractive imaging. Our focus is on information-theoretic fundamental limits under an approximate recovery criterion, considering both discrete and Gaussian models for the sparse non-zero entries, along with Gaussian measurement matrices. In both cases, our bounds provide sharp thresholds with near-matching constant factors in several scaling regimes on the sparsity and signal-to-noise ratio. As a key step towards obtaining these results, we develop new concentration bounds for the conditional information content of log-concave random variables, which may be of independent interest.