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

当信号被假设为稀疏时，即，它可以用从信号分量字典中选择的少量贡献项生成时，遥感或其他类型的信号采集和通信系统中的许多挑战变得可行。该项目解决了建立稀疏恢复信号的通用保证的需要，这些信号与字典的数学结构以及用于合成信号的几何条件有关。这些结果将用于开发从稀疏生成的信号进行准确机器学习预测的要求和恢复保证，检测通过城市网络传播的流行病中出现的热点，以及检测分子动力学中的对称性，以减少计算结合能等各种量时的相关数据。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。 该项目的预期成果包括以下与数据科学广泛相关的目标。第一个是精确恢复的信号，稀疏合成在一个有限或无限维的再生核空间从嘈杂的测量。稀疏恢复是支持向量回归的核心部分，它将在高维空间中对径向高斯核进行。从稀疏恢复的这些结果，预计给洞察模型参数的选择，如高斯的宽度取决于样本的间距。类似的恢复保证也将建立图上的函数，当字典由热核组成时，该热核由一对顶点和热半群下的核的扩散时间索引，也将建立。当传染病在地球仪上传播时，这些结果对于热点的检测具有相关性，这是由当地指数增长和人口中心之间的扩散驱动的。另一个目标是从向量集合的轨道的噪声观测中识别群对称性。这个对称性鉴定问题的动机是在量子化学中的一个应用，在这个应用中，鉴定分子的对称性可以减少根据电子密度估计分子构型的能量或力场所需的样品空间。这个奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.