CIF: Medium: Collaborative Research: Estimating simultaneously structured models: from phase retrieval to network coding

在现代数据密集型科学和工程中，研究人员面临着估计模型的问题，其中可用的观测值远远小于要估计的模型的维数。压缩感知、矩阵补全和其他这类问题成功的关键是正确利用关于模型“结构”的知识。虽然稀疏性等结构已经被单独研究，但“同时结构”的问题被忽略了，因为从业者隐含地假设，简单地将每个结构的已知结果结合起来就可以解决联合问题。有趣的是，PI最近证明，这种方法可能会导致显着的gap.This提案将开发理论和计算上易于处理的方法，估计同时结构化模型与最小的观测。它结合了（1）一个自上而下的方法来理解的基本限制的基础上的几何结构如何相互作用，（2）一个特定的问题，自下而上的方法来利用领域知识，在构建适当的处罚。这项工作解决了各种应用，包括（1）稀疏主成分分析，一个中心问题，在统计寻求近似，但稀疏特征向量，（2）稀疏相位检索和二次压缩传感信号处理，以及（3）通信和网络编码的代码设计。从数据中系统地导出结构化模型的能力将远远超过在大数据和无处不在的计算时代，对工程挑战产生影响。处理具有多种结构的模型提出了深刻的理论和计算挑战，这一建议的重点。在机器学习，信号处理和网络编码的应用进行了讨论。PI将在他们的教学中纳入研究成果，组织技术研讨会，汇集数学家和工程师，并通过夏季研究计划寻求本科生参与这项工作。

In modern data-intensive science and engineering, researchers are faced with estimating models where available observations are far fewer than the dimension of the model to be estimated. The key to the success of compressed sensing, matrix completion, and other problems of this type, is to properly exploit knowledge about the "structure" of the model. While structures such as sparsity have been separately studied, the problem of "simultaneous structures" has been neglected, since it is implicitly assumed by practitioners that simply combining known results for each structure would solve the joint problem. Interestingly, the PIs recently proved that this approach can result in a significant gap.This proposal will develop theory and computationally tractable methods for estimating simultaneously structured models with minimal observations. It combines (1) a top-down approach to understand the fundamental limitations based on the geometry of how structures interact, and (2) a problem-specific, bottom-up approach to exploit domain knowledge in constructing appropriate penalties. This work addresses a variety of applications including (1) sparse principal component analysis, a central problem in statistics seeking approximate but sparse eigenvectors, (2) sparse phase retrieval and quadratic compressed sensing in signal processing, and (3) code design for communications and network coding.The ability to systematically derive structured models from data will have far-reaching impact on engineering challenges in the era of Big Data and ubiquitous computing. Handling models with multiple structures poses deep theoretical and computational challenges that this proposal focuses on. Applications in machine learning, signal processing, and network coding are discussed. The PIs will incorporate research results in their teaching, organize technical workshops to bring together mathematicians and engineers, and seek the involvement of undergraduate students in this work through summer research programs.