# Support Recovery in the Phase Retrieval Model: Information-Theoretic Fundamental Limits

## TL;DR

This paper investigates the fundamental limits of support recovery in phase retrieval models with noisy measurements, providing sharp thresholds and new concentration bounds for information content.

## Contribution

It introduces sharp information-theoretic bounds for support recovery in phase retrieval, including new concentration bounds for log-concave random variables.

## Key findings

- Sharp thresholds for support recovery in phase retrieval models.
- New concentration bounds for the conditional information content.
- Near-matching constants in various sparsity and noise regimes.

## Abstract

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.