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

这项提议的目的是分析现有的方法，并开发新的方法，用于多流形学习任务，其中假设数据由低维结构组成。主要重点将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，以完成探测点云中低维物体的任务；以及分析流行的流形学习算法，特别是在对异常值的稳健性方面。将具体讨论一些应用，例如，运动分割、运动结构、人脸图像的分类、扩散张量图像的分割以及天体物理学中的宇宙学模型的表征。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，其中典型的轨迹通过连续的帧定义曲线；以及也出现在医学成像中，例如在血管网络的检查中。另一个例子是对包含丝状和片状结构的星系分布的研究。众所周知，传统方法在这种情况下是无效的，在过去的十年中，有大量的研究旨在改进这些经典的工具。已经提出了许多多流形建模的方法，主要是由计算机科学家和工程师提出的。应用数学家和统计学家有不同的观点可以提供，他们的贡献是必要的，不仅是在设计新的算法方面，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在要求的。这项提案中的研究将解决这两个问题，发展严格的数学理论，结合精心设计的数值战略，解决具体应用，如运动分割、运动结构、人脸图像分类、医学成像以及天体物理学中宇宙学模型的表征。PIS将通过出版物和软件分享他们的发现，所有这些都可以在网上向科学界和工程界提供。

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.