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

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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.