Robust learning with low-dimensional structure: theory,algorithms and applications

Abstract

In this thesis, we study the robust learning of low-dimensional structures when there are uncertainties in the data. In particular, we consider two structures that are common in real problems: ?low-rank subspace model? that underlies matrix completion and Robust PCA, and ?union-of-subspace model? that arises in the problem of subspace clustering. In the upcoming chapters, we will present (i) stability of matrix factorization and its consequences in the robustness of collaborative filtering (movie recommendations) against manipulators; (ii) sparse subspace clustering under random and deterministic noise; (iii) simultaneous low-rank and sparse regularization for subspace clustering; and (iv) Alternating Robust Subspace Minimization (ARSuMi), a robust matrix recovery algorithm that handles simultaneous noise, missing data and gross corruptions. The results in these chapters either solve a real engineering problem or provide interesting insights into why certain empirically strong algorithms succeed in practice. While in each chapter, only one or two real applications are described and demonstrated, the ideas and techniques in this thesis are general and applicable to any problems having the assumed structures.