Collaborative Research: Multi-manifold data modeling: theory, algorithms and applications

该提案的目的是分析现有方法并开发新方法，用于多流形学习任务，其中假设数据由低维结构组成。 主要重点是研究光谱方法对嵌入高维的低维表面进行聚类和建模的潜力；设计新的基于光谱的方法来检测点云中低维物体的任务；以及流行的流形学习算法的分析，特别是在对异常值的鲁棒性方面。 将具体讨论许多应用，例如运动分割、运动结构、面部图像分类、扩散张量图像分割以及天体物理学中宇宙模型的表征。现代高维数据集通常表现出低维结构。这种情况出现在图像处理中；例如，在目标跟踪中，典型的轨迹定义了通过连续帧的曲线；以及医学成像，例如血管网络的检查。另一个例子是对星系分布的研究，其中包含丝状和片状结构。 众所周知，传统方法在这种情况下是无效的，过去十年已经有大量研究旨在改进这些经典工具。 大多数计算机科学家和工程师已经提出了多种多流形建模方法。 应用数学家和统计学家可以提供不同的观点，并且需要他们的贡献，不仅在设计新算法方面，而且（也许尤其是）在提供理论基础方面，这是该领域的研究人员一直要求的。该提案中的研究将解决这两个问题，发展严格的数学理论，并结合精心设计的数值策略来解决特定应用，例如运动分割、运动结构、面部图像分类、医学成像和天体物理学中宇宙模型的表征。 PI 将通过出版物和软件分享他们的发现，所有这些都可以在线提供给科学和工程界。

The object of this proposal is the analysis of existing methods, and the development of new ones, for the task of multi-manifold learning, where the data is assumed to be comprised of low-dimensional structures. The main focus will be in studying the potential of spectral methods for clustering and modeling of low-dimensional surfaces embedded in high dimensions; in designing new spectral-based approached to the task of detection of low-dimensional objects in point clouds; and the analysis of popular manifold learning algorithms, especially in terms of robustness to outliers. A number of applications will be specifically addressed, for example, motion segmentation, structure from motion, classification of face images, segmentation of diffusion tensor images and the characterization of cosmological models in astrophysics. Modern high-dimensional datasets often exhibit low-dimensional structures. Such situations arise in image processing; e.g., in target tracking, where a typical trajectory defines a curve through successive frames; and also in medical imaging, e.g., in the examination of vascular networks. The study of the galaxy distribution, which contains filamentary and sheet-like structures, is another example. Traditional methods are known to be ineffective in this context and the last decade has seen a massive amount of research aiming at improving on these classical tools. A number of approaches for multi-manifold modeling have been suggested, mostly by computer scientists and engineers. Applied mathematicians and statisticians have different perspectives to offer and their contribution is needed, not only in designing new algorithms but also (and perhaps especially) in providing theoretical foundations, which researchers in the field have been asking for. The research in this proposal will address both issues, developing rigorous mathematical theory combined with carefully designed numerical strategies addressing specific applications, such as motion segmentation, structure from motion, classification of face images, medical imaging and the characterization of cosmological models in astrophysics. The PIs will share their findings through publications and software, all available online to the scientific and engineering communities.