Frames as dictionaries in inverse problems: Recovery guarantees for structured sparsity, unstructured environments, and symmetry-group identification

逆问题中的框架作为字典：结构化稀疏性、非结构化环境和对称群识别的恢复保证

• 批准号: 2308152
• 负责人: Bernhard Bodmann
• 金额: $ 26.67万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2308152&HistoricalAwards=false
• 关键词: Frames; dictionaries; inverse; problems; Recovery

Many challenges in remote sensing or other types of signal acquisition and communication systems become feasible when the signal is assumed to be sparse, that is, it can be generated with a small number of contributing terms selected from a dictionary of signal components. This project addresses the need for establishing universal guarantees for sparse recovery of signals that are related to both the mathematical structure of the dictionary as well as the geometric conditions that are used to synthesize the signal. These results will be used for developing requirements and recovery guarantees for accurate machine learning predictions from a sparsely generated signals, detecting emerging hot spots in an epidemic spreading through a network of cities, and detecting symmetries in molecular dynamics to reduce the relevant data when calculating various quantities such as binding energies. The project also involves the training of graduate students in the mathematical, computational, and interdisciplinary aspects of this project. The expected outcomes of the project include the following goals with broad relevance in data science. The first is the accurate recovery of signals that are sparsely synthesized in a finite or infinite-dimensional reproducing kernel space from noisy measurements. Sparse recovery is a central part of support vector regression, which will be carried out for radial Gaussian kernels in high-dimensional spaces. These results from sparse recovery are expected to give insight in the choice of model parameters such as the width of the Gaussian depending on the spacing of the samples. Similar recovery guarantees will also be established for functions on graphs, when the dictionary consists of heat kernels that are indexed by the pair of a vertex and a time for the diffusion of the kernel under the heat semigroup, will also be established. These results have relevance for the detection of hot spots when an infectious disease spreads across the globe, driven by local exponential growth and diffusion between population centers. Another goal is to identify group symmetries from noisy observations of the orbit of a collection of vectors. This question of symmetry identification is motivated by an application in quantum chemistry where identifying symmetries of molecules can reduce the space of samples needed to estimate energies or force fields for molecular configurations based on electron densities.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.

当假定信号稀疏时，遥感或其他类型的信号采集和通信系统的许多挑战变得可行，也就是说，可以使用少数从信号组件字典中选择的术语来生成。该项目解决了建立通用保证的需求，以稀疏地恢复与字典的数学结构以及用于合成信号的几何条件相关的信号。这些结果将用于开发需求和恢复保证，以确保从稀疏产生的信号中进行准确的机器学习预测，从而在通过城市网络中检测出流行病扩散中的新斑点，并在分子动力学中检测到对称性，以减少相关数据时计算各种各种数据。诸如结合能等数量。该项目还涉及该项目的数学，计算和跨学科方面的研究生培训。 该项目的预期结果包括以下目标在数据科学中具有广泛相关性。首先是从噪声测量值中的有限或无限二维复制空间中稀疏合成的信号的准确恢复。稀疏恢复是支撑矢量回归的中心部分，将在高维空间中针对径向高斯内核进行。稀疏恢复的这些结果预计将洞悉模型参数的选择，例如高斯的宽度，具体取决于样品的间距。当字典由由顶点对索引的热核组成时，还将建立类似的恢复保证金，以在图形上的功能上建立类似的恢复保证。当局部指数增长和人口中心之间的扩散驱动时，当传染病在全球范围内传播时，这些结果与检测热点有关。另一个目标是从对矢量集合轨道的嘈杂观察中识别群体对称性。这个对称性识别的问题是由量子化学中的应用激发的，在量子化学中，识别分子的对称性可以减少基于电子密度估计能量或力场所需的样品的空间。通过基金会的智力优点和更广泛的影响评估标准通过评估来支持。