Estimating Low Dimensional, High-Density Structure

高维空间中的数据现在非常普遍。这个项目将开发分析这些高维数据的方法。这些数据可能包含隐藏的结构。例如，星团（具有大量点的小区域）可以像弦一样伸展，形成一种称为细丝的结构。各个领域的科学家都需要对这些物体进行定位。这是具有挑战性的，因为数据通常非常嘈杂。该项目将开发严格合理和计算有效的方法来提取此类结构。这些方法将应用于天体物理学、地震学、生物学和神经科学等领域的一系列问题。该项目将推进几个领域的知识，包括计算几何、天文学、机器学习和统计学。发现隐藏结构对科学发现和降维是有帮助的。目前关于非线性降维的许多理论都假定隐藏结构是光滑流形，并且具有很强的限制性。数据可能集中在一个低维但非常复杂的集合附近，例如相交流形的并集。现有的算法，如子空间约束Mean Shift在路口附近表现出不稳定的行为。该项目将针对这些情况开发改进的算法。与此同时，当代理论在这些情况下崩溃，本项目将发展新的理论来解决上述问题。本文提出了一种将不同维数的点云分解成子集的完整方法（称为奇异聚类）。

Data in high dimensional spaces are now very common. This project will develop methods for analyzing these high dimensional data. Such data may contain hidden structures. For example, clusters (which are small regions with a large number of points) can be stretched out like a string forming a structure called a filament. Scientists in a variety of fields need to locate these objects. It is challenging since the data are often very noisy. This project will develop rigorously justified and computationally efficient methods for extracting such structures. The methods will be applied to a diverse set of problems in astrophysics, seismology, biology, and neuroscience. The project will advance knowledge in several fields including computational geometry, astronomy, machine learning, and statistics.Finding hidden structure is useful for scientific discovery and dimension reduction. Much of the current theory on nonlinear dimension reduction assumes that the hidden structure is a smooth manifold and is very restrictive. The data might be concentrated near a low dimensional but very complicated set, such as a union of intersecting manifolds. Existing algorithms, such as the Subspace Constrained Mean Shift exhibit erratic behavior near intersections. This project will develop improved algorithms for these cases. At the same time, contemporary theory breaks down in these cases and this project will develop new theory to address the aforementioned problem. A complete method (which will be called singular clusters) will be developed for decomposing point clouds of varying dimensions into subsets.