[Paper Review] Phase Retrieval for Sparse Signals

Yang Wang, Zhiqiang Xu|arXiv (Cornell University)|Oct 3, 2013

Advanced X-ray Imaging Techniques3 citations

TL;DR

This paper establishes the theoretical foundations for recovering sparse signals from magnitude-only measurements, proving that 2k generic real vectors and 4k−2 generic complex vectors suffice for k-sparse phase retrieval. It introduces a null space property for ℓ₁ minimization that is both necessary and sufficient for successful recovery, providing the first such condition for sparse phase retrieval in the literature.

ABSTRACT

The aim of this paper is to build up the theoretical framework for the recovery of sparse signals from the magnitude of the measurement. We first investigate the minimal number of measurements for the success of the recovery of sparse signals without the phase information. We completely settle the minimality question for the real case and give a lower bound for the complex case. We then study the recovery performance of the $\ell_1$ minimization. In particular, we present the null space property which, to our knowledge, is the first sufficient and necessary condition for the success of $\ell_1$ minimization for $k$-sparse phase retrievable.

Motivation & Objective

• To determine the minimal number of measurements required for successful recovery of k-sparse signals from magnitude-only measurements.
• To establish a theoretical framework for ℓ₁ minimization in sparse phase retrieval, particularly its conditions for success.
• To derive a null space property that is both necessary and sufficient for ℓ₁ minimization to recover k-sparse signals in phase retrieval.
• To resolve the minimality question for k-sparse phase retrieval in the real case and provide a tight lower bound for the complex case.

Proposed method

• Prove that for real-valued signals, m ≥ 2k generic measurements are both necessary and sufficient for k-sparse phase retrieval.
• Establish that in the complex case, m ≥ 4k−2 generic vectors are sufficient for k-sparse phase retrieval, and conjecture this bound is tight.
• Introduce a null space property for ℓ₁ minimization in phase retrieval, generalizing known results from compressive sensing.
• Use frame matrix analysis and equivalence class structures to characterize phase retrievability via partition-based conditions on null spaces.
• Prove equivalence between phase retrievability and the nonexistence of certain nontrivial null vectors satisfying specific ratio conditions across partitions of measurements.
• Leverage unimodular equivalence and complex vector space structure to derive necessary and sufficient conditions for injectivity of the magnitude map on sparse equivalence classes.

Experimental results

Research questions

• RQ1 What is the minimal number of measurements required to uniquely recover a k-sparse signal from magnitude-only measurements in the real case?
• RQ2 What is the minimal number of measurements required for k-sparse phase retrieval in the complex case, and is 4k−2 the optimal bound?
• RQ3 What conditions on the measurement matrix ensure that ℓ₁ minimization successfully recovers a k-sparse signal from magnitude data?
• RQ4 Can a null space property be formulated that is both necessary and sufficient for ℓ₁ minimization to succeed in sparse phase retrieval?
• RQ5 How do partition-based conditions on null vectors relate to the injectivity of the magnitude measurement map on sparse equivalence classes?

Key findings

• For real-valued signals, m ≥ 2k generic measurements are both necessary and sufficient for k-sparse phase retrieval, and this bound is sharp.
• In the complex case, m ≥ 4k−2 generic vectors are sufficient for k-sparse phase retrieval, and the authors conjecture this bound is tight.
• The paper presents the first known null space property that is both necessary and sufficient for ℓ₁ minimization to succeed in k-sparse phase retrieval over real and complex domains.
• The null space property is formulated via the nonexistence of nontrivial null vectors satisfying specific ratio conditions across partitions of the measurement indices.
• The equivalence between phase retrievability and the absence of such structured null vectors is rigorously proven for both real and complex cases.
• The results show that phase retrieval for k-sparse signals requires significantly fewer measurements than full-dimension phase retrieval, reducing the requirement from 4d−4 in the complex case to 4k−2 when k ≪ d.