Deep Learning on Manifolds: New Architectures and Theoretical Foundations

Over the last couple of decades, deep learning approaches have achieved breakthrough performance in a broad range of learning problems from a variety of applications fields such as image recognition, speech recognition, natural language processing and others. Deep learning has also served as the main impetus for the advancement of recent artificial intelligence (AI) technologies.

The main goal of this project is to develop new deep neural network (DNN) architectures, computational algorithms and foundational theory for deep learning in complex domains where the data are complex geometric objects. The underlying geometry of the space will be utilized and exploited for developing the DNNs. Such development will enable the application of deep learning models in medical imaging, computer vision, computer graphics, recommender systems and neuroscience to learning problems where the inputs are complex images, diffusion matrices, shapes or other geometric objects.

In addition, results of the project will be integrated in a course on Geometry and Statistics for undergraduate and graduate students. The principal investigators will also be involved in the research training of students, possibly students from under-represented groups.

The main goals of the research program are to develop general deep neural network architectures on manifolds and take some major steps toward understanding their theoretical foundations. Specifically, this project will (1) develop extrinsic deep neural networks (eDNNs) on manifolds by generalizing the popular feedforward neural networks in the Euclidean space to manifolds utilizing equivariant embeddings; (2) develop intrinsic deep neural networks (iDNNs) on manifolds employing a Riemannian structure of the manifold; (3) develop general retraction-based convolutional neural networks (RCNNs) on manifolds; (4) characterize theoretical properties of eDNNs, iDNNs and RCNNs on manifolds by studying their approximation properties as well as the estimation error for a class of empirical risk minimizers over the DNN class on manifolds; and (5) develop user-friendly software packages that can be readily used by the practitioners.

The research program explores the interface among geometry, statistics and machine learning, and aims to broaden the paradigm of deep learning by extending geometric deep learning to manifolds.

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.