Frame mechanics: Dynamical principles for optimal redundant expansions

框架力学的概念解决了构建大量最优冗余、稳定的框架扩展的需求，这已经成为遥感或无线传输、模数转换（如音频和视频编码）、基于分组的网络通信、噪声不敏感的量子计算以及最近的压缩感知中的数学应用的核心。尽管它很受欢迎，但对近最佳帧的搜索主要是在小维度上成功的，或者它必须依赖于特定的组表示属性，或者使用随机化原则。在框架力学中，研究者正在研究一种替代传统的、结构化的或随机的设计方法，通过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。一般目标是找到(1)合适的框架动力学，(2)合适的初始化，并获得(3)逼近误差的确定性控制。该项目的预期结果包括利用最近建立的等角紧框架结构的数值结果来验证Zauner猜想（在所有有限维希尔伯特空间中存在最大Gabor框架），为无线或基于分组的网络通信中的损失不敏感传输构建Grassmannian框架和融合框架的控制近似。并设计了基于量子混沌动力学的压缩感知矩阵，提高了感知矩阵的受限等距特性。冗余信号表示的数学称为框架理论。就实际用途而言，帧是在存储、传输或接收数据时包含或删除重复信息的工具。帧在现代技术的许多数据密集型领域变得至关重要，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，帧的优化设计的进步已经被硬件产生的数据的快速增长所超越。在框架力学中，研究者和他的学生探索了一种全新的策略来克服这个问题：构建这种最佳框架的负担放在计算机上，让框架以一种驱使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保证在以前的尝试失败的情况下找到许多最优帧。框架力学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保持稳定。

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.