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

信号处理中的应用需要对噪声和数据丢失具有健壮性的有效信号表示法。因此，信号经常相对于称为帧的冗余字典来表示。帧是应用数学、计算机科学和工程中的标准工具，并在无线通信、编码和语音识别等应用领域找到了用途。特殊的框架系列专为特定的信号处理应用而设计，以提供最佳的效率和稳健性。具有规定数据的帧的集合形成一组复杂的矩阵，称为帧空间。帧空间的基本特征没有被很好地理解，这意味着即使是简单的问题，关于在帧之间进行内插的可能性，或者关于随机帧具有良好性质的概率，也仍然没有得到解决。在这个项目中，这些问题是通过辛几何的镜头来看待的，辛几何是一个植根于经典力学的领域，旨在探索框架理论中出现的各种对称性。这个项目将应用辛几何中的技术来对框架空间的几何结构进行新的洞察，提供解决这些长期存在的问题的新的理论结果，以及生成均匀随机框架的实用算法，以供更广泛的框架理论和信号处理社区使用。这个项目的一个重要组成部分是通过互动演示将框架理论和辛几何的思想介绍给新的受众，从学龄儿童和爱好者通过互动演示，到研究生通过正式的研究培训，到非专家数学家和工程师通过调查辛几何的实际应用的说明性写作。本项目将应用辛几何的工具来解决框架理论中的三个具体的公开问题。该项目的第一个目标是使用辛技术来推导概率保证，即从给定的框架空间(例如，单位范数紧框架空间)随机抽取的框架具有期望的性质，例如来自压缩感知的受限等距性质。推动这一项目的关键见解是辛几何为框架空间提供了一个新的坐标系，具有方便的测度论性质。这一观察结果也对该项目的第二个目标有影响，该项目的第二个目标是开发有效采样帧空间的新算法。框架空间由于其复杂的几何和拓扑结构，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族有效的马尔可夫链算法来采样框架。这些算法将提供实用的好处，作为一种工具，用于试验性地探索帧的统计数据，例如部分帧算子的特征值分布，并用于为压缩传感应用生成随机帧。该项目的第三个目标是将这些辛技术扩展到处理广义帧，包括融合帧和算子值帧，在这种情况下提供概率保证和采样算法，并应用于具有块稀疏结构的信号的压缩传感。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.