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

Massive high-dimensional data are ubiquitous in many scientific and engineering disciplines, such as bioinformatics, computer vision, neuroimaging, and signal processing. This proposal is motivated by emerging tools for analyzing data from these disciplines, such as nonsmooth, manifold-based learning with high-dimensional and multidimensional data. Building on the synergy among statistics, machine learning, and optimization, this research will focus on the development of new optimization algorithms and theory for nonsmooth manifold optimization. The project will also build on existing optimization strengths to develop new methods and theory in statistics and machine learning. Software packages will be developed to make the research outcomes readily available to other researchers and practitioners. In addition, the project will enhance the future technical workforce through the training of graduate students. 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.