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.
The projects supported by this grant center on developing new methodology, and analyzing new and/or existing methods, for multi-manifold learning and computational geometry. 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 galaxy distributions, which are comprised of filaments and sheets, is another example. From this comes the modeling hypothesis that the data lie near manifolds (e.g., curves or surfaces) of comparatively low dimensionality. This sort of modeling assumption happens to be necessary to avoid the so-called `curse of dimensionality'. This assumption is common a wide array of applications, such as motion segmentation, wearable action recognition, medical imaging and model inference in cosmology. Traditional methods are ineffective in multi-manifold learning, and the last decade has seen a massive amount of research aiming at improving on these classical tools, making multi-manifold learning one of the most important research trends in machine learning in the last 15 years or so. A large number of approaches 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 contributes to this effort. Indeed, a number of popular methods for multi-manifold learning are rigorously analyzed in some of the projects supported by this grant. New methods are also suggested as part of these projects. The other thrust is in the closely related field of computational geometry, and estimation of geometric characteristics, which is another thriving field of research. In some areas of science and engineering, researchers are taking a much more geometrical approach to modeling data, beyond functional representations, which fail in high-dimensions. In such situations, geometric characteristics become central. This is for example the case in astrophysics, where topological quantities are now commonly used to study the distribution of galaxies in space. Some projects in this proposal focus on studying some geometric characteristics of points clouds (which is often how data is presented to the scientist) and their relation to their continuous counterpart. For a simple example, one project involves the estimation of the perimeter of a set based on a sample of points from that set. The research described in this proposal is part of a truly multidisciplinary effort, involving many applied mathematicians, statisticians, computer scientists, and engineers, as well as other scientists from other fields dealing with particular kinds of data. In that line of work, our research projects provide much needed theoretical foundations for multi-manifold modeling and computational geometry. The products resulting from this research include software and research articles, all available online to the scientific community and public at large. Besides both applied and theoretical research, the proposal also contains an educational component. The PI has developed a graduate course at his home institution incorporating recent research in multi-manifold learning. And one of the projects will be the center piece of a graduate student's doctoral thesis.
