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

University Of Houston, Houston TX

Investigators

Abstract

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.