Collaborative Research: New Methods, Theory and Applications for Nonsmooth Manifold-Based Learning

Nowadays, the availability of massive data is continuously increasing, primarily due to the continued advancement of technology. As a consequence, massive high-dimensional data are ubiquitous in many scientific and engineering disciplines, such as bioinformatics, computer vision, neuroimaging, and signal processing. The nonsmooth manifold-based learning with high-dimensional and multidimensional data is in general complicated due to its intrinsic non-convexity and non-smoothness. This project will address both statistical and computational issues of nonsmooth manifold-based learning and explore its new applications. It is known that statistical modeling of high-dimensional data may include the non-smooth regularization in the objective function, and some may even involve non-convex manifold constraints such as orthogonality constraints. The manifold-based learning offers a powerful framework for dimension reduction and signal processing. The combination of non-smooth regularization and non-convex manifold constraints brings new opportunities and challenges for designing optimization algorithms with convergence guarantees and also for developing new statistical methods and theory. The research outcomes of this project will provide new powerful analytic tools in nonsmooth manifold-based learning with theoretical guarantees. Software packages will be developed to make the research outcomes readily available to other researchers and practitioners. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.