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

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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.