CIF:Small:Next-Generation Compressive Phase-Retrieval Using Sparse-Graph Codes: Theory, Design and Applications

本研究解决了测量系统设计和大规模结构化信号恢复的基础科学，当(a)关键相位信息无法测量，(b)可用的测量很少。在技术文献中也被称为压缩相位检索，这项研究已应用于许多重要的科学领域，如光学，量子物理，生物医学成像，天文学和材料科学。无法测量相位信息使得信号恢复问题特别具有挑战性。大多数先前的方法要么是计算上的限制，因此难以扩展，要么是基于未经验证的启发式，没有性能保证。相比之下，本研究从根本上解决了规模的挑战，并在压缩相位检索系统的理论、设计、实验评估和应用中提供了可证明的性能保证。本研究解决了下一代压缩相位检索系统的理论和算法基础。这些都来源于一种新的跨学科混合工具，包括编码理论、图论、统计信号处理，以及它们与光学成像和量子信息系统的应用集成。该研究围绕建立在一组稀疏图代码上的新范式展开，这些代码代表了对基于计算密集型凸松弛方法或没有性能保证的贪婪启发式的现有方法的彻底背离。这项研究有可能彻底改变下一代压缩相位检索系统的设计和应用，类似于低密度奇偶校验（LDPC）码如何彻底改变现代通信系统，在以下方面：(1)测量成本（接近基本极限）；（ii）计算和内存成本（实现实时或近实时处理）；（iii）可证明的性能保证（关于问题规模和对噪声的鲁棒性）。

This research addresses the fundamental science of measurement system design and recovery of large-scale structured signals of interest when (a) critical phase information cannot be measured, and (b) very few measurements are available. Also known in the technical literature as compressed phase-retrieval, this research has applications to a plethora of important scientific fields like optics, quantum physics, bio-medical imaging, astronomy, and material science. The inability to measure phase information renders the signal recovery problem particularly challenging. Most prior approaches are either computationally prohibitive and therefore hard to scale, or based on unproven heuristics that come with no performance guarantees. In contrast, this research fundamentally addresses the challenge of scale together with provable performance guarantees in the theory, design, experimental evaluation, and applications of compressed phase-retrieval systems. This research addresses the theoretical and algorithmic foundations for next-generation compressed phase-retrieval systems. These are derived from a novel interdisciplinary mix of tools from coding theory, graph theory, statistical signal processing, as well as their integration into applications involving optical imaging and quantum information systems. The research revolves around a novel paradigm built on a family of sparse-graph codes that represent a radical departure from existing approaches based on either computationally intensive convex relaxation methods or greedy heuristics without performance guarantees. This research has the potential to revolutionize the design and applications of next-generation compressed phase retrieval systems, similar to how Low-Density-Parity-Check (LDPC) codes have revolutionized modern communication systems, with respect to (i) measurement cost (approaching the fundamental limits); (ii) computational and memory cost (enabling real-time or near-real-time processing); and (iii) provable performance guarantees (with respect to the problem scale and robustness to noise).