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，其中该方法不像现有方法那样保持原始空间中的局部距离，而是保持原始空间中的密度。其动机是双重的：利用黎曼几何的结果，研究人员已经证明，通常不可能保持距离，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，可以用密度来最好地正式描述，可以认为这种方法直接保留了感兴趣的实际信息。这是通过产生最小二乘问题的新公式来实现的，如前期工作所示，或者是待开发的凸优化问题。其次，该项目开发了以非参数方式估计子流形上的点的密度的理论和方法，这是整体方法的第一步。这包括渐近结果，它依赖于子流形的维度，而不是当电流存在时环境空间的维度。这与流行的结论形成了鲜明对比，即高维空间中的非参数估计很难处理。将进行理论、方法和实验的发展。各种高维数据，如通常编码的文本文件、图像或天文光谱，已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解决这些问题。这些数据对国土安全、医学、环境遥感、电子商务和许多其他领域至关重要。这项工作的学术价值在于引入了一种新的方法来制定和分析这类数据的两个基本统计运算，称为降维和密度估计。其中每一项都可能为非常需要的高维统计领域开辟新的途径。这项工作的更广泛影响是，分析员能够可靠地识别高维数据中的离群值和星团--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物体。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，以最大限度地影响数据分析的许多领域。

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.