Collaborative Research: Applications of Symplectic Geometry to Frame Theory and Signal Processing

信号处理中的应用需要有效的信号表示，这些信号表示对噪声和数据丢失具有鲁棒性。因此，信号经常相对于称为帧的冗余字典来表示。框架是应用数学、计算机科学和工程中的标准工具，并且已经在诸如无线通信、编码和语音识别的应用领域中找到了用途。特殊的帧系列是为特定的信号处理应用而设计的，以提供最佳的效率和鲁棒性。具有规定数据的帧的集合形成称为帧空间的复杂矩阵集。框架空间的基本特征还没有得到很好的理解，这意味着即使是关于框架之间插值的可能性或关于随机框架具有良好性质的概率的听起来很简单的问题仍然没有解决。在这个项目中，这些问题是通过辛几何的透镜来看待的，辛几何是一个起源于经典力学的领域，旨在利用框架理论中出现的各种对称性。该项目将应用辛几何的技术，对框架空间的几何结构给予新的认识，提供新的理论结果，解决这些长期存在的问题，以及产生均匀随机框架的实用算法，供更广泛的框架理论和信号处理社区使用。该项目的一个重要组成部分是向新受众介绍框架理论和辛几何的思想，从学龄学生和业余爱好者通过互动演示，到研究生通过正式的研究培训，非-专家数学家和工程师通过简要的写作调查辛几何的实际应用。这个项目将应用辛几何的工具，以解决三个具体的主要开放框架理论的问题。该项目的第一个目标是使用辛技术来获得概率保证，即从给定的帧空间（例如，单位范数紧帧的空间）随机绘制的帧具有期望的属性，例如来自压缩感知的受限等距属性。推动这个项目的关键见解是，辛几何提供了一个新的坐标系的框架空间与方便的测量理论的性质。这一观察结果也对该项目的第二个目标产生了影响，即开发新的算法来有效地对帧空间进行采样。框架空间由于其复杂的几何结构和拓扑结构，很难用直接方法进行采样，但新的辛框架空间坐标将导致一个有效的马尔可夫链框架采样算法族。这些算法将提供实际的好处，作为一种工具，实验探索统计的帧，如特征值分布的部分帧运营商和压缩感知应用程序生成随机帧。该项目的第三个目标是扩展这些辛技术来处理广义框架，包括融合框架和算子值框架，在此设置中提供概率保证和采样算法，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响进行评估来支持审查标准。

Applications in signal processing require efficient signal representations which are robust to noise and data loss. Signals are therefore frequently represented with respect to a redundant dictionary called a frame. Frames are standard tools in applied mathematics, computer science, and engineering, and have found uses in application domains such as wireless communication, coding, and speech recognition. Special families of frames are designed for specific signal processing applications to provide optimal efficiency and robustness. The collection of frames with prescribed data forms a complicated set of matrices called a frame space. Basic features of frame spaces are not well understood, which means even simple-sounding questions about the possibility of interpolating between frames or about the probability that a random frame has good properties remain unsolved. In this project these questions are viewed through the lens of symplectic geometry, a field with roots in classical mechanics which is designed to exploit the sorts of symmetries that arise in frame theory. This project will apply techniques from symplectic geometry to give new insight into the geometric structure of frame spaces, providing new theoretical results resolving these longstanding questions as well as practical algorithms for generating uniformly random frames for use by the broader frame theory and signal processing communities. A significant component of this project is to introduce ideas from frame theory and symplectic geometry to new audiences, from school-age students and hobbyists through interactive demonstrations, to graduate students through formal research training, to non-expert mathematicians and engineers through expository writing surveying the practical applications of symplectic geometry.This project will apply tools from symplectic geometry to address three specific major open problems in frame theory. The first aim of the project is to use symplectic techniques to derive probabilistic guarantees that a frame drawn randomly from a given frame space (for example, the space of unit-norm tight frames) enjoys desirable properties, such as the Restricted Isometry Property from compressed sensing. The key insight driving this project is that symplectic geometry provides a new coordinate system for frame spaces with convenient measure theoretic properties. This observation also has implications for the second aim of the project, which is to develop novel algorithms for efficiently sampling frame spaces. Frame spaces are inherently hard to sample with direct methods because of their complicated geometry and topology, but the new symplectic frame space coordinates will lead to an efficient family of Markov chain algorithms for sampling frames. These algorithms will provide practical benefits as a tool for experimentally exploring statistics of frames such as eigenvalue distributions of partial frame operators and for generating random frames for compressed sensing applications. The third aim of the project is to extend these symplectic techniques to handle generalized frames, including fusion frames and operator-valued frames, providing probabilistic guarantees and sampling algorithms in this setting, with applications to compressed sensing of signals with block sparse structure.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.