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

Robust Recovery of Subspace Structures by Low-Rank Representation

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed Low-Rank Representation (LRR), which seeks the lowest-rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.Comment: IEEE Trans. Pattern Analysis and Machine Intelligenc

Progressive Attenuation of the Longitudinal Kinetics in the Common Carotid Artery: Preliminary in Vivo Assessment

Longitudinal kinetics (LOKI) of the arterial wall consists of the shearing motion of the intima-media complex over the adventitia layer in the direction parallel to the blood flow during the cardiac cycle. The aim of this study was to investigate the local variability of LOKI amplitude along the length of the vessel. By use of a previously validated motion-estimation framework, 35 in vivo longitudinal B-mode ultrasound cine loops of healthy common carotid arteries were analyzed. Results indicated that LOKI amplitude is progressively attenuated along the length of the artery, as it is larger in regions located on the proximal side of the image (i.e., toward the heart) and smaller in regions located on the distal side of the image (i.e., toward the head), with an average attenuation coefficient of −2.5 ± 2.0%/mm. Reported for the first time in this study, this phenomenon is likely to be of great importance in improving understanding of atherosclerosis mechanisms, and has the potential to be a novel index of arterial stiffness