Abstract

In this dissertation we discuss three problems characterized by hidden structure or information. The first part of this thesis focuses on extracting subspace structures from data. Subspace Clustering is the problem of finding a multi-subspace represen-tation that best fits a collection of points taken from a high-dimensional space. As with most clustering problems, popular techniques for subspace clustering are often difficult to analyze theoretically as they are often non-convex in nature. Theoret-ical analysis of these algorithms becomes even more challenging in the presence of noise and missing data. We introduce a collection of subspace clustering algorithms, which are tractable and provably robust to various forms of data imperfections. We further illustrate our methods with numerical experiments on a wide variety of data segmentation problems. In the second part of the thesis, we consider the problem of recovering the seem-ingly hidden phase of an object from intensity-only measurements, a problem which naturally appears in X-ray crystallography and related disciplines. We formulate the