There is a growing interest in computer science, engineering, and mathematics for modeling signals in terms of union of subspaces and manifolds. Subspace segmentation and clustering of high-dimensional data drawn from a union of subspaces are especially important with many practical applications in computer vision, image and signal processing, communications, and information theory. For example, recent paradigms for reconstructing signals assume models that consist of union of subspaces, e.g., the reconstruction of signals with finite rates of innovation. Another example is in compressed sampling where signals are assumed to be sparse in some basis or in some dictionary. This assumption implies that the signals live in a union of subspaces. The subspace clustering problem in computer vision is yet another important example in which data are drawn from a union of low-dimensional subspaces. Thus, a mathematical framework for finding such models from observed data is fundamental. In this project the investigator develops a mathematical framework together with algorithms for data modeling in terms of union of subspaces and manifolds. The mathematical theory is connected to the geometry of Hilbert and Banach spaces, topology, nonlinear approximation, optimization, and probability.
The investigator develops a mathematical framework and algorithms for describing data by representing them as composed of components that live in restricted sets of the data space, for instance, in subspaces or manifolds. The theory and methods developed by the investigator unify, extend, and complement some of the techniques used in sampling theory, subspace clustering, and the dictionary design problem. The applications are fundamental to many problems in engineering and biomedicine, including motion tracking in videos, data classification and segmentation such as face recognition, and brain morphology. The project is supported by the Division of Mathematical Sciences and the Division of Computing and Communication Foundations.