With the advance of modern technology and computing power, processing and analyzing of data in three and higher dimensions becomes a ubiquitous task in diverse fields such as medical imaging, computational chemistry, computational biology, social networks and many others. For many problems in practice, data is commonly associated with certain coherent and nonlinear structure. Mathematically, this allows us to model data sets as points sampled on manifolds usually of low dimensions embedded in a high dimensional ambient space. Different from image and signal processing which handle functions on flat domains with well-developed tools for processing and learning, manifold-structured data is far more challenging due to their complicated geometry. For example, the same geometric object can take very different coordinate representations due to the variety of embeddings, transformations or representations (imagine the same human body shape can have different poses as its nearly isometric embedding ambiguities). These ambiguities form an infinite dimensional isometric group and make higher-level tasks in manifold-structured data analysis and learning even more challenging. To overcome these challenges, it becomes increasingly important to develop new tools in both theoretical and computational point of views for processing manifold structured data.
This project proposes to investigate analyzing and learning of manifold-structured data by bridging connection from geometric partial differential equations (PDEs) and learning theory to intrinsic data analysis. The major objectives of this project contain three components. The first part is to investigate a framework of geometric-PDEs-based methods to a data structure for manifolds represented as incomplete inter-point distance. The second part is to overcome the challenge of poor performance using intrinsic descriptors to handling not nearly isometric manifolds. In the third part, a new method of defining geometric convolution on manifolds is considered. This provides a building block of constructing the proposed geometric convolutional neural network for conducting deep learning on manifold-structured data. By collaborating with biomedical engineers, applications such as human brain mappings will also be explored. The new methodologies and research findings resulting from the proposed work will lead to new ways of tackling problems in manifold-structured data analysis and will be integrated into my future teaching and course projects in appropriate ways. The education plan is to provide unique opportunities to train undergraduate and graduate students interested in exploring geometry and learning on manifold-structured data, and to reach out the general public.
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.