Multi-Manifold Data Modeling and Applications

Many common data sets can be modeled by a mixture of simple geometric objects, e.g., manifolds. A well known example of such data is the NIST images of hand-written digits. The effective modeling of such data together with its careful analysis is a challenging mathematical problem. Earlier work at the beginning of the current century revealed effective methods for modeling data by a single manifold. Later work modeled data by an arrangement of affine subspaces. The generalization of these works to multi-manifold data modeling is currently being developed. It has many important applications, for example, in motion segmentation, hybrid representation of images, classification of face images and classification of hand-written digits.

Different approaches for this problem have utilized theoretical insights from various mathematical disciplines and also inspired new theoretical observations. Those mathematical disciplines include probabilistic modeling, algebraic geometry, information theory, computational topology and geometry, statistical inference, linear and multilinear algebra, numerical analysis, differential geometry, and spectral graph theory. This general problem is also closely related to other directions in current applied mathematics, in particular, sparse approximation.