Frame mechanics: Dynamical principles for optimal redundant expansions

BodmannDMS-1109545帧机制的概念解决了利用帧构建大量最佳冗余、稳定扩展的需求，帧已成为数学在遥感或无线传输、音频和视频编码等模数转换、基于分组的网络通信、噪声不敏感的量子计算以及最近也在压缩传感中的应用的核心。尽管它很受欢迎，但寻找近乎最佳的帧主要是在小维度上成功的，或者它必须依赖于特定的群表示属性，或者使用随机化原则。在框架力学中，研究人员正在研究一种替代传统、结构化或随机设计方法的方法，即让框架在驱动它们走向最优的流动下进化，而不是直接构建它们。总的目标是找到(1)适当的帧动力学，(2)适当的初始化，以及(3)逼近误差的确定性控制。该项目的预期成果包括利用最近建立的关于构造等角紧框架的数值结果来验证Zauner猜想(所有有限维Hilbert空间中存在最大Gabor框架)，为无线或基于分组的网络通信中的不敏感传输构造Grassman框架和融合框架的受控近似，以及基于量子混沌动力学设计用于压缩感知的矩阵，从而改善感知矩阵的受限等距性质。冗余信号表示的数学称为框架理论。出于实际目的，帧是在存储、传输或接收数据时合并或删除重复信息的工具。在现代技术的许多数据密集型领域，帧已经变得至关重要，因为重复的信息有助于补偿传输设备和传感器的错误。然而，在过去的几十年里，我们的硬件产生的数据的快速增长已经超过了帧的最佳设计的进展。在框架力学中，研究人员和他的学生探索了一种全新的策略来克服这个问题：构建这种最优框架的负担被放在计算机上，计算机让框架以一种驱动它们走向最优的方式进化。这个项目的目标是证明这种动态设计策略在数学上是有保证的，可以找到许多先前尝试失败的最佳帧。帧结构使我们能够最大限度地提高遥感、地震和医学成像、无线和光纤通信的性能，并使互联网传输在网络中断时变得稳健。

BodmannDMS-1109545 The concept of frame mechanics addresses the need for constructing an abundance of optimal redundant, stable expansions with frames, which have become central to applications of mathematics in remote sensing or wireless transmissions, in analog-digital conversion such as audio and video encoding, in packet-based network communications, noise-insensitive quantum computing and recently also in compressive sensing. Despite its popularity, the search for near-optimal frames has been successful mostly in small dimensions, or it had to rely on specific group-representation properties, or the use of randomization principles. In frame mechanics, the investigator is studying an alternative to the conventional, structured or random design methods by letting frames evolve under flows which drive them towards optimality, instead of constructing them directly. The general objectives are to find (1) appropriate frame dynamics, (2) suitable initializations, and to obtain (3) deterministic control of the approximation error. The envisioned outcome of the project includes leveraging recently established numerical results on the construction of equiangular tight frames for the verification of Zauner's conjecture (the existence of maximal Gabor frames in all finite-dimensional Hilbert spaces), constructing controlled approximations of Grassmannian frames and fusion frames for loss-insensitive transmissions in wireless or packet-based network communications, and the design of matrices for compressive sensing based on quantum chaotic dynamics which improve the restricted isometry properties of sensing matrices. The mathematics of redundant signal representations is called frame theory. For practical purposes, a frame is a tool which incorporates or removes repetitive information when data is stored, transmitted or received. Frames have become essential in many data-intensive areas of modern technology, because the repetitive information helps compensate errors of transmission devices and sensors. However, over the last decades, progress in the optimal design of frames has been outpaced by the rapid growth of data generated by our hardware. In frame mechanics, the investigator and his students explore a fundamentally new strategy to overcome this problem: The burden of constructing such optimal frames is put on the computer, which lets frames evolve in a way that drives them towards optimality. The goal of this project is to demonstrate that this dynamic design strategy is mathematically guaranteed to find many optimal frames where previous attempts failed. Frame mechanics allows us to maximize performance in remote sensing, seismic and medical imaging, wireless and fiber-optic communications, and to make internet transmissions robust to network outages.