Many signals of interest in data analysis have lower-dimensional structure than their ambient dimension suggests. Exploiting this latent structure in sampling and reconstruction strategies can dramatically increase algorithmic robustness to both noise and missing data. The theory of compressed sensing shows that if a signal of interest is sparse --- that is, well-approximated by some small subset of a dictionary of basis elements, then the signal can be acquired from a reduced number of measurements and reconstructed using efficient convex programming techniques. However, current compressed sensing theory is limited largely to finite-dimensional, well-conditioned, and uniformly bounded dictionaries, and these restrictions limit the scope of applications. Using notions such as variable-density and weighted sparsity, the investigator and her colleagues aim to develop a range of structure-dependent sampling theorems that merge finite-dimensional sparsity constraints with infinite-dimensional smoothness constraints, and which naturally extend the compressed sensing methodology to infinite-dimensional and unbounded function systems. As a related goal of this proposal, the investigator has teamed up with professors in computer science and electrical engineering at UT Austin to develop an interdisciplinary statistical signal processing seminar series. The investigator has also recently established the first Association for Women in Mathematics student chapter at UT Austin in order to foster the advancement of women in mathematics.
Broadly speaking, this proposal puts forth theoretical tools that can be used to design strategies for acquiring high-dimensional data as efficiently as possible, given any known lower-dimensional structure of the data and within the framework of the acquisition process at hand. Magnetic Resonance Imaging (MRI) is one driving application. Here, one would like to reduce the MRI scan time as much as possible while still acquiring a clear image of the brain, neck, or other internal structure. By exploiting the underlying structure of natural images, such as the localization of information content of the image to boundaries between different materials in the brain, MRI scanning technology can become significantly cheaper and faster, and preliminary experiments by the investigator and collaborators suggest that the sampling strategies put forth in this proposal have the potential to speed up MRI scan timestenfold. Uncertainty Quantification is another application of the proposed research. Here, one is interested in analyzing the sensitivity of high-dimensional nonlinear models to small changes in input parameters. Applications range from the design of civil infrastructure to be robust in the face of extreme climate, to the assessment of the stability for climate models with respect to perturbations in initial weather conditions. Generally speaking, Uncertainty Quantification involves repeatedly perturbing the initial conditions of the model at hand, simulating the model at each of these perturbations, and analyzing the resulting output statistics. Since such simulations are expensive for high-dimensional nonlinear models, one would like to derive strategies for simulating perturbed input parameters so as to gain as much information about the model from as few simulations as possible.
