Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional subspace that can change over time, albeit gradually. The problem of tracking such data (and the subspaces) while being robust to outliers is called robust subspace tracking (RST). This article provides a magazine-style overview of the entire field of robust subspace learning and tracking. In particular solutions for three problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition (S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an entire data vector is either an outlier or an inlier. The S+LR formulation instead assumes that outliers occur on only a few data vector indices and hence are well modeled as sparse corruptions.Comment: To appear, IEEE Signal Processing Magazine, July 201

Compressed Sensing in Resource-Constrained Environments: From Sensing Mechanism Design to Recovery Algorithms

Compressed Sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for reconstruction. It is promising that CS can be utilized in environments where the signal acquisition process is extremely difficult or costly, e.g., a resource-constrained environment like the smartphone platform, or a band-limited environment like visual sensor network (VSNs). There are several challenges to perform sensing due to the characteristic of these platforms, including, for example, needing active user involvement, computational and storage limitations and lower transmission capabilities. This dissertation focuses on the study of CS in resource-constrained environments. First, we try to solve the problem on how to design sensing mechanisms that could better adapt to the resource-limited smartphone platform. We propose the compressed phone sensing (CPS) framework where two challenging issues are studied, the energy drainage issue due to continuous sensing which may impede the normal functionality of the smartphones and the requirement of active user inputs for data collection that may place a high burden on the user. Second, we propose a CS reconstruction algorithm to be used in VSNs for recovery of frames/images. An efficient algorithm, NonLocal Douglas-Rachford (NLDR), is developed. NLDR takes advantage of self-similarity in images using nonlocal means (NL) filtering. We further formulate the nonlocal estimation as the low-rank matrix approximation problem and solve the constrained optimization problem using Douglas-Rachford splitting method. Third, we extend the NLDR algorithm to surveillance video processing in VSNs and propose recursive Low-rank and Sparse estimation through Douglas-Rachford splitting (rLSDR) method for recovery of the video frame into a low-rank background component and sparse component that corresponds to the moving object. The spatial and temporal low-rank features of the video frame, e.g., the nonlocal similar patches within the single video frame and the low-rank background component residing in multiple frames, are successfully exploited