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

首席研究员和他的合作者的目标是开发足够简单的有效数据建模范例，以进行统计推断。目前的科学研究以及工业应用都产生并依赖于海量的、高维的、可能被破坏的数据集。应用数学家和统计学家在这一领域的一个主要关注点是定量几何数据建模。为了有效地分析大量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。该提案为这种有效的几何模型提供了数学范例。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值策略，解决具体和重要的应用。尽管最近在这一领域取得了进展，但仍有许多开放的方向，本研究项目解决了其中的几个方向。更具体地说，该提案侧重于几何数据建模的几个重要方向。一个方向旨在解决单一健壮的子空间建模中与新的学习和计算范式有关的现代问题，这些新范式迄今在这一背景下几乎没有得到解决。另一个方向将探索通过具有新范式和新视角的多个子空间或流形来建模数据的重要问题。该提案还将强调低阶和稀疏建模的具体范例，这些范例是由重要应用引起的，例如用于目标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.