Understanding Data Through Mappings

通过映射理解数据

• 批准号: 1228304
• 负责人: Leonidas Guibas
• 金额: $ 78.5万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1228304&HistoricalAwards=false
• 关键词: Understanding; Data; Through; Mappings

The investigators will study the problem of computing informative, structure-preserving mappings between related data sets, and especially data sets of a geometric character, such as images, videos, GPS traces, 3D scans, or microarray data. Unlike classical data fusion where the goal is to find exact correspondences, the main emphasis here is to be able to map data at different levels of abstraction and to incorporate uncertainty and ambiguity directly into the map formulation. This raises challenging issues, both at the representational and the algorithmic levels. The aim is to develop efficient multi-resolution techniques through which data set relationships can be compactly encoded, compared, etc. --- making data relationships into precise, tangible objects that can be explored, just like the data themselves. The work will involve tools from a wide variety of mathematical disciplines, including aspects of differential geometry and topology, functional and harmonic analysis, algebraic and computational topology, machine learning and statistics, discrete and continuous optimization, scientific computing, and finally discrete algorithms and data structures. Across all human activities, from science and engineering to medicine, commerce, and defense, massive data sets are becoming more and more available and more and more crucial to improved efficiency and enhanced functionality. As our data sets grow in size and number, they become increasingly interconnected and inter-related. This is because data is captured about the same or related entities in the physical or virtual words (e.g., different images of the same building, different logs of the same user), as well as because the data itself reflects an underlying reality that has symmetries, regularities, and other shared structure. Thus it makes sense to analyze data sets jointly, exploiting this shared structure to do individual operations on data sets better. Through the above mapping machinery it will be possible to organize data collections into (possibly overlapping) groups of related sets or parts thereof, separating what is common from what is variable within each group and across groups, and understanding the main axes of variability.

研究人员将研究计算相关数据集之间的信息性、结构保持映射的问题，特别是几何特征的数据集，如图像、视频、GPS轨迹、3D扫描或微阵列数据。与目标是找到精确对应的经典数据融合不同，这里的主要重点是能够绘制不同抽象级别的数据，并将不确定性和多义性直接纳入地图公式中。这提出了具有挑战性的问题，无论是在代表性层面还是在算法层面。其目的是开发高效的多分辨率技术，通过这些技术可以对数据集关系进行紧凑的编码、比较等-使数据关系成为可以探索的精确、有形的对象，就像数据本身一样。这项工作将涉及各种数学学科的工具，包括微分几何和拓扑学、泛函和调和分析、代数和计算拓扑学、机器学习和统计、离散和连续优化、科学计算，最后是离散算法和数据结构。在所有人类活动中，从科学和工程到医学、商业和国防，海量数据集正变得越来越可用，对提高效率和增强功能越来越重要。随着我们的数据集在大小和数量上的增长，它们变得越来越相互联系和相互关联。这是因为在物理或虚拟单词(例如，同一建筑物的不同图像、同一用户的不同日志)中捕获关于相同或相关实体的数据，以及因为数据本身反映了具有对称性、规律性和其他共享结构的底层现实。因此，联合分析数据集，利用这种共享结构更好地对数据集进行单独的操作是有意义的。通过上述绘图机制，将有可能将数据收集组织成相关集合或其部分的(可能重叠的)组，将每个组内和组间的共同之处与可变之处分开，并理解可变性的主轴。