Sparse and low-rank structures in robust principal component analysis, compressed sensing with corruptions, and phase retrieval

Placeholder Show Content


In this dissertation, we discuss three related problems: robust principal component analysis, compressed sensing with corruptions, and signal recovery from quadratic measurements. Robust principal component analysis (RPCA) is a computational framework that aims to decompose a matrix as the sum of a low-rank component and a sparse component. We prove that under mild technical assumptions, a tractable convex optimization works for an exact decomposition. In the case in which there are missing entries, we propose another convex optimization and show that a low-rank matrix can be exactly recovered from a small portion of entries with a constant proportion of corruptions. Further, this dissertation improves some existing results in compressed sensing with grossly corrupted measurements in the literature. For both the Gaussian and non-Gaussian sensing matrices, we derive theorems showing that by weighted L1 minimization, a sparse signal can be recovered from a few linear observations with constant proportion of corruptions. Finally, we discuss the problem of signal recovery from intensity measurements. For general signal recovery, we improve existing results in the literature, showing that the signal can be recovered exactly up to a global phase factor provided the number of measurements is in the order of the dimension of the signal. For sparse signal recovery, we derive both necessary and sufficient conditions for the number of Gaussian measurements, such that a natural convex relaxation works for the recovery of the signal.


Type of resource text
Form electronic; electronic resource; remote
Extent 1 online resource.
Publication date 2013
Issuance monographic
Language English


Associated with Li, Xiaodong, 1985-
Associated with Stanford University, Department of Mathematics.
Primary advisor Candès, Emmanuel J. (Emmanuel Jean)
Thesis advisor Candès, Emmanuel J. (Emmanuel Jean)
Thesis advisor Papanicolaou, George
Thesis advisor Ye, Yinyu
Advisor Papanicolaou, George
Advisor Ye, Yinyu


Genre Theses

Bibliographic information

Statement of responsibility Xiaodong Li.
Note Submitted to the Department of Mathematics.
Thesis Thesis (Ph.D.)--Stanford University, 2013.
Location electronic resource

Access conditions

© 2013 by Xiaodong Li
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

Also listed in

Loading usage metrics...