Union of Subspaces and Manifold Data Modeling: Theory, Algorithms, Testing, and Applications

AldroubiDMS-1108631 在计算机科学、工程学和数学中，人们越来越感兴趣的是用子空间和流形的联合来建模信号。 从子空间的并集中提取的高维数据的子空间分割和聚类对于计算机视觉、图像和信号处理、通信和信息理论中的许多实际应用尤其重要。 例如，用于重构信号的最近范例假设由子空间的并集组成的模型，例如，用有限的新息率重建信号。 另一个例子是在压缩采样中，其中信号被假设为在某个基或某个字典中是稀疏的。 这个假设意味着信号存在于子空间的并集中。 计算机视觉中的子空间聚类问题是另一个重要的例子，其中数据是从低维子空间的并集中提取的。 因此，从观测数据中找到这种模型的数学框架是至关重要的。 在这个项目中，研究人员开发了一个数学框架，以及子空间和流形的联合数据建模算法。 数学理论与希尔伯特和巴拿赫空间的几何学、拓扑学、非线性逼近、最优化和概率有关。 研究人员开发了一个数学框架和算法，用于描述数据，将它们表示为由数据空间的有限集合中的组件组成，例如子空间或流形。 研究者开发的理论和方法统一，扩展和补充了抽样理论，子空间聚类和字典设计问题中使用的一些技术。 这些应用是工程和生物医学中许多问题的基础，包括视频中的运动跟踪，数据分类和分割，如人脸识别和大脑形态学。 该项目由数学科学部和计算与通信基础部支持。

AldroubiDMS-1108631 There is a growing interest in computer science, engineering, and mathematics for modeling signals in terms of union of subspaces and manifolds. Subspace segmentation and clustering of high-dimensional data drawn from a union of subspaces are especially important with many practical applications in computer vision, image and signal processing, communications, and information theory. For example, recent paradigms for reconstructing signals assume models that consist of union of subspaces, e.g., the reconstruction of signals with finite rates of innovation. Another example is in compressed sampling where signals are assumed to be sparse in some basis or in some dictionary. This assumption implies that the signals live in a union of subspaces. The subspace clustering problem in computer vision is yet another important example in which data are drawn from a union of low-dimensional subspaces. Thus, a mathematical framework for finding such models from observed data is fundamental. In this project the investigator develops a mathematical framework together with algorithms for data modeling in terms of union of subspaces and manifolds. The mathematical theory is connected to the geometry of Hilbert and Banach spaces, topology, nonlinear approximation, optimization, and probability. The investigator develops a mathematical framework and algorithms for describing data by representing them as composed of components that live in restricted sets of the data space, for instance, in subspaces or manifolds. The theory and methods developed by the investigator unify, extend, and complement some of the techniques used in sampling theory, subspace clustering, and the dictionary design problem. The applications are fundamental to many problems in engineering and biomedicine, including motion tracking in videos, data classification and segmentation such as face recognition, and brain morphology. The project is supported by the Division of Mathematical Sciences and the Division of Computing and Communication Foundations.