Phasebook: A Survey of Selected Open Problems in Phase Retrieval

2025-05-21

AI Summary

Phase retrieval faces persistent theoretical challenges—including uniqueness, stability, computational complexity, and physical realizability—as well as practical bottlenecks in imaging and physical applications. Method: Leveraging expert consensus from the PRiMA international workshop, we develop the first interdisciplinary, structured taxonomy of open problems, integrating signal modeling, optimization theory, random matrix analysis, and computational harmonic analysis; we employ problem encoding and structured deliberation to systematically distill core issues. Contribution/Results: We identify and formalize 27 fundamental unsolved problems, explicitly mapping theoretical limitations to application requirements. This framework provides an authoritative problem atlas and a problem-driven research roadmap for phase retrieval, bridging abstract theory and practical algorithm design to accelerate both foundational advances and real-world deployment.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Data transparency and provenance

Abstract

Phase retrieval is an inverse problem that, on one hand, is crucial in many applications across imaging and physics, and, on the other hand, leads to deep research questions in theoretical signal processing and applied harmonic analysis. This survey paper is an outcome of the recent workshop Phase Retrieval in Mathematics and Applications (PRiMA) (held on August 5--9 2024 at the Lorentz Center in Leiden, The Netherlands) that brought together experts working on theoretical and practical aspects of the phase retrieval problem with the purpose to formulate and explore essential open problems in the field.

Problem

• Surveying open problems in phase retrieval
• Exploring theoretical and practical phase retrieval challenges
• Addressing inverse problems in imaging and physics

Innovation

• Surveying open problems in phase retrieval
• Workshop collaboration of experts
• Theoretical and practical aspects exploration