CAREER: Sparsity-aware Sampling Theorems and Applications

职业：稀疏感知采样定理和应用

• 批准号: 1255631
• 负责人: Rachel Ward
• 金额: $ 42万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1255631&HistoricalAwards=false
• 关键词: CAREER; Sparsity; aware; Sampling; Theorems

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.

在数据分析中，许多感兴趣的信号具有比其环境维度所暗示的更低维的结构。 在采样和重建策略中利用这种潜在结构可以显着提高算法对噪声和丢失数据的鲁棒性。 压缩感知理论表明，如果感兴趣的信号是稀疏的-也就是说，很好地近似由基元字典的一些小的子集，那么信号可以从减少数量的测量和重建使用有效的凸规划技术。 然而，目前的压缩感知理论主要局限于有限维、良好条件和一致有界的字典，这些限制限制了压缩感知的应用范围。利用变密度和加权稀疏等概念，研究人员和她的同事旨在开发一系列结构相关的采样定理，将有限维稀疏性约束与无限维平滑性约束合并，并将压缩感知方法自然扩展到无限维和无界函数系统。 作为这项提案的相关目标，研究人员与UT Austin的计算机科学和电气工程教授合作，开发了一个跨学科的统计信号处理研讨会系列。 研究人员最近还成立了第一个协会的妇女在数学学生章在德州奥斯汀，以促进妇女在mathematics.Broadly的进步，这一建议提出了理论工具，可用于设计策略，以获取高维数据尽可能有效地，给定任何已知的低维结构的数据和在手头的收购过程的框架内。磁共振成像（MRI）是一种驱动应用。 这里，人们希望尽可能地减少MRI扫描时间，同时仍然获取大脑、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扫描技术可以变得更加便宜和快速，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扫描时间加快十倍。不确定性量化是所提出的研究的另一个应用。 这里，人们感兴趣的是分析高维非线性模型对输入参数微小变化的敏感性。其应用范围从设计在极端气候下坚固耐用的民用基础设施，到评估气候模型在初始天气条件扰动下的稳定性。一般来说，不确定性量化涉及反复扰动模型的初始条件，在每个扰动下模拟模型，并分析结果输出统计。 由于这样的模拟是昂贵的高维非线性模型，人们希望得到的策略，用于模拟扰动输入参数，以便获得尽可能多的信息，从尽可能少的模拟模型。