Abstract: The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. Instead of directly reconstruct the incomplete distance matrix, we consider a low-rank matrix completion method to reconstruct the associated Gram matrix with respect to a suitable basis. Computationally, simple and fast algorithms are designed to solve the proposed problem. Theoretically, the well known restricted isometry property (RIP) can not be satisfied in the scenario. Instead, a dual basis approach is considered to theoretically analyze the reconstruction problem. Furthermore, by introducing a new condition on the basis called the correlation condition, our theoretical analysis can be also extended to a more general setting to handle low-rank matrix completion problems under any given non-orthogonal basis. This new condition is polynomial time checkable and holds for many cases of deterministic basis where RIP might not hold or is NP-hard to verify. If time permits, I will also discuss a combination of low-rank matrix completion with geometric PDEs on point clouds to understanding manifold-structured data represented as incomplete inter-point distance data.

Bio: Dr. Rongjie Lai received his Ph.D. degree in applied mathematics from the University of California, Los Angeles. He is currently an assistant professor at the Rensselaer polytechnic Institute. Dr. Lai’s research interests are mainly in developing mathematical and computational tools for analyzing and processing signals, images as well as unorganized data using methods of variational partial differential equations, computational differential geometry and learning. In 2018, Dr. Lai was granted an NSF CAREER award for his research in geometry and learning for manifold-structured data in 3D and higher dimension.