Density-Preserving Maps

密度保持贴图

• 批准号: 0907484
• 负责人: Alexander Gray
• 金额: $ 18万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907484&HistoricalAwards=false
• 关键词: Density; Preserving; Maps

This project will investigate two aspects of high-dimensional statistics. First, it develops a new alternative paradigm for nonlinear dimension reduction (often called manifold learning) in which, instead of preserving local distances in the original space as done by existing approaches, the approach preserves the densities in the original space. The motivation is twofold: Using results from Riemannian geometry, the investigators have shown that is not possible in general to preserve distances, and that it is always possible to preserve densities; in addition, because perhaps the common scientific use of nonlinear dimension reduction methods is to visualize clusters and outliers, which are arguably best formally described in terms of densities, it can be argued that this approach directly preserves the actual information of interest. This is achieved by means of novel formulations resulting in least-squares problems, as shown in preliminary work, or convex optimization problems to be developed. Second, the project develops theory and methodology for nonparametrically estimating the densities of points lying on a submanifold, which is needed as the first step in the overall approach. This includes asymptotic results which are dependent on the dimension of the submanifold rather than that of the ambient space, as current exist. This provides contrast to the popular conclusion that nonparametric estimation in high dimensional spaces is simply intractable. Theoretical, methodological, and experimental development will be performed.Very high-dimensional data, such as text documents, images, or astronomical spectra as typically encoded, have become increasingly important and prevalent, while statistical theory and methods have only recently attacked such problems with full vigor. Such data are critical for homeland security, medicine, remote sensing of the environment, e-commerce, and a host of other domains. The intellectual merit of the work is the introduction of a new way of formulating and analyzing two fundamental statistical operations on such data, called dimension reduction and density estimation. Each of these could open the door to new avenues in the much-needed area of very high-dimensional statistics. The broader impact of the work is the transformative ability of analysts to reliably identify outliers and clusters in high-dimensional data -- for example such a tool could help astronomers identify new types of astrophysical objects. The work will be distributed as part of a well-distributed state-of-the-art toolbox of statistical methods to maximize impact across many areas of data analysis.

这个项目将研究高维统计的两个方面。 首先，它开发了一种新的非线性降维（通常称为流形学习）的替代范式，其中，该方法保留了原始空间中的密度，而不是像现有方法那样保留原始空间中的局部距离。 动机有两个方面：利用黎曼几何的结果，研究人员已经表明，一般不可能保持距离，而保持密度总是可能的;此外，由于非线性降维方法的常见科学用途可能是可视化集群和离群值，这可以说是最好的密度形式描述，可以认为，这种方法直接保留了感兴趣的实际信息。 这是通过新的配方导致最小二乘问题，如在初步工作中所示，或凸优化问题的开发。 第二，该项目开发的理论和方法，非参数估计的密度点躺在一个子流形，这是需要作为第一步的整体方法。 这包括渐近结果是依赖于维数的子流形，而不是周围的空间，因为目前存在。 这提供了对比的流行的结论，在高维空间的非参数估计是简单的棘手。 理论，方法和实验的发展将进行。非常高维的数据，如文本文件，图像，或天文光谱作为典型的编码，已经变得越来越重要和流行，而统计理论和方法只是最近才全力攻击这样的问题。 这些数据对于国土安全、医学、环境遥感、电子商务和许多其他领域至关重要。 这项工作的智力价值是引入了一种新的方法来制定和分析两个基本的统计操作，这些数据被称为降维和密度估计。 每一个都可以为非常高维统计这一急需的领域开辟新的途径。 这项工作的更广泛影响是分析人员可靠地识别高维数据中的异常值和集群的变革能力-例如，这样的工具可以帮助天文学家识别新类型的天体物理对象。 这项工作将作为一个分布良好的最先进的统计方法工具箱的一部分进行分发，以最大限度地扩大对许多数据分析领域的影响。