Some problems in geometric data analysis

对数据集的分析越来越几何化。在某些应用中，例如在天体物理学中对宇宙网的分析中，这可以说是最自然的方法。在其他情况下，通过几何结构对数据集建模可以绕过函数的使用，由于所谓的“维度诅咒”，函数通常难以在高维中处理。该项目旨在通过发展新方法和新理论，在几何数据分析的一般领域，特别是在聚类、降维和表面估计等领域做出贡献。数据分析的几何方法已经确立。聚类、降维和流形/曲面估计已经得到了很好的发展，目前正在以鲁棒主成分分析、子空间聚类、流形或曲面聚类、流形学习、几何统计、计算几何等形式进行研究。这种研究的绝大多数是方法论的，或者是应用于特定领域的特定问题，理论基本上是落后的。例如，在子空间聚类、流形嵌入和传感器定位等重要领域就是这种情况。本项目旨在为这些领域提供理论见解。虽然方法论往往远远领先于理论，但良好和及时的理论分析可以为应用问题提供一些启示，并且有时可以为更有效的方法论设计提供信息。而这种方法在几何统计的某些领域是缺失的。因此，该项目还包括开发实用的方法，可以证明与手头问题的最小最大性能界限相匹配，特别是在流形估计和分布几何特征估计领域。

The analysis of datasets is increasingly geometrical. In some applications, such as in the analysis of the cosmic web in astrophysics, this is arguably the most natural approach. In others, modeling datasets via geometric structures allows to bypass the use of functions, which are in general difficult to deal with in high-dimensions because of the so-called "curse of dimensionality". The project aims at making contributions in this general area of geometrical data analysis, and in particular in fields like clustering, dimensionality reduction, and surface estimation, via the development of new methodology and new theory.Geometrical approaches to data analysis are well-established. Clustering, dimensionality reduction, and manifold/surface estimation, are well-developed, with ongoing work in the form of robust PCA, subspace clustering, manifold or surface clustering, manifold learning, geometric statistics, computational geometry, etc. The vast majority of this research is methodological or applied to a particular problem in a specific field, and theory is by and large lagging behind. This is, for example, the case in important areas such as subspace clustering, manifold embedding and sensor localization. This project has the ambition to contribute theoretical insights in those areas. While methodology tends to be well ahead of theory, good and timely theoretical analyses can shed some light on applied problems, and can sometimes inform the design of more effective methodology. And such methodology is missing in some areas of geometric statistics. Thus the project also includes the development of practical methodology that provably matches the minimax performance bounds available for the problem at hand, particularly in the areas of manifold estimation and the estimation of geometric characteristics of a distribution.