CAREER: New Paradigms in Geometric Analysis of Data Sets and their Applications

PI和他的合作者将开发算法，用于从大量高维数据集中检测和恢复潜在的稀疏几何结构。特别是，他们计划探索以下框架：用于检测点云内低维几何结构的几何优化；有效检测局部尺度的多尺度方法及其组合以获取最相关的局部和全局几何信息；在线和自适应算法组织数据作为流形的混合物，同时分离异常值。所提出的方法将通过对性能的理论保证来证明，这些方法将应用于各种数据集，其中许多数据集将由工业合作者提供。应用包括：视频监控摄像机中运动物体的自动检测；视频图像的运动分割，通过动态CT扫描将大脑中的血管自动分割为动脉和静脉。最近，对某些类型的数据集（如数字卫星图像和磁共振图像）的分析和操作发生了根本性的转变。这种革命依赖于这样一个事实：虽然这些图像表面上具有复杂的高维结构，但实际上它们是相对低维的或“稀疏”的。基本的观察是，这种稀疏性可以被用来更快地获取、传输、重建和分析这些图像。PI和他的合作者正在将这种“降维”技术扩展到更一般的数据集实例，目的是识别看似高维的数据集合实际上更简单，然后掌握简化的结构是什么。这类研究有几个重要的应用，涉及对数据进行计算机辅助决策，具有安全和医学意义。希望这项研究能够产生快速、有效和经过验证的算法，用于分离随时间变化的数据的各种重要特征。这项研究的实际好处将包括安全摄像头的可靠自动化。本提案中建议的许多应用程序和主题都可供广泛的社区访问。PI计划利用这种可访问性，以便将研究工作与年轻研究人员和学生的教育结合起来。特别是，PI致力于为各级数学教育工作者提供材料，并让本科生和研究生参与新兴工业研究。PI将通过出版物和软件分享他的共同发现，所有这些都可以在线提供给科学和工程社区以及广大公众。

The PI and his collaborators will develop algorithms for detecting and recovering underlying sparse geometric structures from massive high-dimensional data sets. In particular, they plan to explore the following frameworks: geometric optimization for the purpose of detecting low-dimensional geometric structures within point clouds; multiscale methods for the effective detection of local scales and their combination for capturing the most relevant local and global geometric information; online and adaptive algorithms for organizing data as mixtures of manifolds while separating outliers. The proposed methodologies will be justified by theoretical guarantees on performance, and these methodologies will be applied to a variety of data sets, many of which will be provided by industrial collaborators. The applications include: automatic detection of moving objects in video surveillance cameras; motion segmentation of video images, automatic segmentation of blood vessels in the brain taken via dynamic CT scans into arteries and veins. Recently there has been a fundamental shift in the analysis and manipulation of certain types of data sets such as digital satellite images and magnetic resonance images (MRI). This revolution relies on the fact that while such images seemingly have a complex and high-dimensional structure, in fact they are relatively low-dimensional or "sparse". The basic observation was that this sparsity could be exploited to more rapidly acquire, transmit, reconstruct, and analyze such images. The PI and his collaborators are extending such "dimensionality reduction" techniques to more general instances of data sets with the aim of identifying when seemingly high-dimensional collections of data are actually much more simple, and to then get a grip on what the simplified structure is. Such research has several important applications related to making computer aided decisions about data which has both security and medical significance. The hope is that the research yields speedy, efficient, and proven algorithms for separating various and important features of data which is changing in time. The practical benefits of the research would include reliable automation of security cameras. Many of the applications and themes suggested in this proposal are accessible to a broad community. The PI plans to take advantage of this accessibility in order to integrate the research effort with the education of younger researchers and students. In particular, the PI is committed to provide material to mathematics educators at all levels and involve undergraduate and graduate students in emerging industrial research. The PI will share his joint findings through publications and software, all available online to the scientific and engineering communities as well as the public at large.