CAREER: Sparsity-aware Sampling Theorems and Applications

数据分析中的许多感兴趣的信号具有比其周围维度所暗示的更低的维度结构。在采样和重建策略中利用这种潜在结构可以显著提高算法对噪声和丢失数据的稳健性。压缩感知理论表明，如果感兴趣的信号是稀疏的-即，由基本元素词典的某个小子集很好地逼近，则可以从减少的测量次数中获取信号，并使用有效的凸规划技术进行重建。然而，目前的压缩感知理论主要局限于有限维的、条件良好的、一致有界的字典，这些限制限制了应用范围。利用变密度和加权稀疏性等概念，这位研究人员和她的同事们致力于开发一系列结构相关的采样定理，这些定理将有限维稀疏约束和无限维光滑约束结合在一起，自然地将压缩传感方法扩展到无限维和无界函数系统。作为这项提议的一个相关目标，研究人员与德克萨斯大学奥斯汀分校的计算机科学和电气工程教授合作，开发了一系列跨学科的统计信号处理研讨会。研究人员最近还在德克萨斯大学奥斯汀分校建立了第一个妇女数学学生协会，以促进妇女在数学领域的进步。总的来说，这项建议提出了理论工具，可用于设计策略，尽可能有效地获取高维数据，给定任何已知的低维数据结构，并在手头获取过程的框架内。磁共振成像(MRI)是一种驱动力应用。在这里，人们希望尽可能地减少MRI扫描时间，同时仍然获得大脑、颈部或其他内部结构的清晰图像。通过利用自然图像的底层结构，例如将图像的信息内容定位到大脑中不同材料之间的边界，MRI扫描技术可以变得明显更便宜和更快，研究人员和合作者的初步实验表明，该提案中提出的采样策略有可能将MRI扫描时间缩短为原来的1倍。不确定性量化是本研究的另一个应用。这里，人们感兴趣的是分析高维非线性模型对输入参数的微小变化的敏感性。应用范围从设计民用基础设施以使其在面对极端气候时保持稳健，到评估气候模型相对于初始天气条件的扰动的稳定性。一般来说，不确定性量化包括反复扰动手头模型的初始条件，在每个扰动下模拟模型，并分析结果的输出统计。由于这样的模拟对于高维非线性模型来说是昂贵的，人们想要推导出用于模拟扰动输入参数的策略，以便从尽可能少的模拟中获得关于模型的尽可能多的信息。

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 limitedlargely 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.