Density-Preserving Maps

这个项目将研究高维统计的两个方面。首先，它为非线性降维（通常称为流形学习）开发了一种新的替代范式，在这种范式中，该方法不像现有方法那样保留原始空间中的局部距离，而是保留原始空间中的密度。动机是双重的：利用黎曼几何的结果，研究人员已经表明，一般来说，保持距离是不可能的，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聚类和离群值，这可能是密度方面最好的正式描述，因此可以认为这种方法直接保留了感兴趣的实际信息。这是通过产生最小二乘问题的新公式来实现的，如初步工作中所示，或有待开发的凸优化问题。其次，该项目开发了非参数估计子流形上点密度的理论和方法，这是整个方法的第一步。这包括依赖于子流形的维数而不是当前存在的环境空间维数的渐近结果。这与普遍认为高维空间中的非参数估计难以处理的结论形成了对比。将进行理论、方法和实验开发。非常高维的数据，如典型编码的文本文件、图像或天文光谱，已经变得越来越重要和普遍，而统计理论和方法直到最近才全力以赴地研究这些问题。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在于引入了一种新的方法来阐述和分析这类数据的两种基本统计操作，即降维和密度估计。其中每一项都可以在非常需要的高维统计领域打开通往新途径的大门。这项工作更广泛的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性能力——例如，这样的工具可以帮助天文学家识别新的天体物理对象类型。这项工作将作为分布良好的最先进的统计方法工具箱的一部分分发，以最大限度地影响许多数据分析领域。

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.