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

TL;DR

This paper provides a comprehensive overview of robust subspace learning and tracking, focusing on methods that handle outliers in PCA, subspace estimation, and dynamic data scenarios.

Contribution

It reviews and discusses solutions for robust PCA, robust subspace tracking, and robust subspace recovery, highlighting the sparse+low-rank matrix decomposition approach.

Findings

Robust PCA effectively separates low-rank structure from sparse outliers.

Robust subspace tracking adapts to changing subspaces in data streams.

Robust subspace recovery identifies outliers as entire data vectors or sparse corruptions.

Abstract

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.