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主要关注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。该建议为这种有效的几何模型提供了数学范式。它计划为这些范式发展严格的数学理论，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，但仍有许多开放的方向，其中一些是本研究项目所涉及的。更具体地说，该建议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鲁棒子空间建模中的现代问题，涉及到迄今为止在此设置中几乎没有解决的学习和计算的新范式。另一个方向将探索用新的范例和视角通过多个子空间或流形建模数据的重要问题。该提案还将强调低秩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.