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

当信号被假定为稀疏时，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，也就是说，可以用从信号分量词典中选择的少量贡献项来生成信号。该项目涉及建立稀疏恢复信号的普遍保障的需要，这些保障既与词典的数学结构有关，也与合成信号所用的几何条件有关。这些结果将用于开发从稀疏生成的信号进行准确的机器学习预测的需求和恢复保证，检测通过城市网络传播的流行病中的新热点，以及在计算结合能等各种量时检测分子动力学中的对称性以减少相关数据。该项目还包括在该项目的数学、计算和跨学科方面对研究生进行培训。该项目的预期成果包括以下在数据科学方面具有广泛相关性的目标。第一个是从噪声测量中准确地恢复在有限或无限维再生核空间中稀疏合成的信号。稀疏恢复是支持向量回归的核心部分，将对高维空间中的径向高斯核进行稀疏恢复。稀疏恢复的这些结果有望为模型参数的选择提供洞察力，例如依赖于样本间距的高斯宽度。对于图上的函数，当字典由以顶点对为索引的热核组成时，也将建立类似的恢复保证，并且还将建立热半群下核扩散的时间。这些结果对于检测传染病在全球范围内传播时的热点具有相关性，这种传播是由当地人口中心之间的指数增长和扩散驱动的。另一个目标是从一组向量的轨道的噪声观测中识别群对称性。这个对称性识别问题的动机是量子化学中的一个应用，在这个应用中，识别分子的对称性可以减少基于电子密度估计分子配置的能量或力场所需的样本空间。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.