Novel Paradigms in Geometric Modeling of Large and High-Dimensional Data Sets

主要研究者和他的合作者旨在开发有效的数据建模范式，这些范式对于统计推断来说足够简单。当前的科学研究以及工业应用产生并依赖于大量、高维和可能损坏的数据集。应用数学家和统计学家在这一领域的一个主要重点是定量几何数据建模。为了有效地分析大数据并获得有意义的统计推断，底层几何模型需要足够简单。该提案提出了这种有效的几何模型的数学范式。它计划为这些范例开发严格的数学理论，并结合精心设计的数字策略来解决具体和重要的应用。尽管最近在这一领域取得了进展，有许多开放的方向，其中一些研究项目地址。更具体地说，该提案集中在几何数据建模的几个重要方向。一个方向的目的是解决现代问题，在单一的强大的子空间建模的学习和计算的新范式，几乎没有解决到目前为止，在这个设置。另一个方向将探索通过多个子空间或流形以新的范式和视角建模数据的重要问题。该提案还将强调低秩和稀疏建模的特定范例，这些范例由重要应用引起，例如用于对象识别的近似最近子空间，改进的特征跟踪，计算机视觉中的运动结构，以及大气科学中的稀疏建模。

The principal investigator and his collaborators aim to develop effective data modeling paradigms that are sufficiently simple for statistical inference. Current scientific investigations, as well as industrial applications, produce and rely on massive, high-dimensional and possibly corrupted data sets. A major focus of applied mathematicians and statisticians in this area has been on quantitative geometric data modeling. In order to effectively analyze large data and obtain meaningful statistical inference, the underlying geometric models need to be sufficiently simple. The proposal suggests mathematical paradigms for such effective geometric models. It plans to develop rigorous mathematical theory for these paradigms combined with carefully designed numerical strategies addressing specific and important applications. Despite the recent progress in this area, there are many open directions, several of which this research project addresses.More specifically, the proposal focuses on several important directions of geometric data modeling. One direction aims to address modern issues in single robust subspace modeling with respect to new paradigms of learning and computation that have hardly been addressed so far in this setting. Another direction will explore important issues in modeling data by multiple subspaces or manifolds with new paradigms and perspectives. The proposal will also emphasize specific paradigms of low-rank and sparse modeling, which are induced by important applications, such as approximate nearest subspace for object recognition, improved feature tracking, structure from motion in computer vision, and sparse modeling in the atmospheric sciences.