Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery (1711.09492v4)

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.

• The paper introduces robust PCA techniques that separate low-rank structures from outliers, mitigating classical PCA’s sensitivity to noise.
• It extends the discussion to robust subspace tracking through online frameworks like ReProCS, enabling adaptive management of dynamic data.
• The comparative analysis on video datasets demonstrates improved efficiency and accuracy, validating the practicality of these robust subspace methods.

Insights on Robust Subspace Learning: A Detailed Analysis

The paper "Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery" by Namrata Vaswani, Thierry Bouwmans, Sajid Javed, and Praneeth Narayanamurthy provides an exhaustive overview of the vast domain of robust subspace learning, with particular emphasis on methods of robust Principal Component Analysis (RPCA), robust subspace tracking (RST), and robust subspace recovery (RSR).

This paper begins by addressing a fundamental issue with classical PCA techniques, which are notably sensitive to outliers. In real-world scenarios, data is often corrupted by noise, discontinuities, or outages. The discussion encapsulates both static and dynamic problems and presents a succinct comparative analysis of several robust techniques in tackling these scenarios, primarily through the robust PCA framework.

Key Concepts and Techniques

1. Robust PCA (RPCA):
   • RPCA has been formulated to address the core problem of estimating a low-rank subspace when outliers perturb the observed data. The authors thoroughly explore solutions such as Principal Component Pursuit (PCP), and non-convex alternatives like AltProj and RPCA-GD, which provide promising results with reduced computational complexity. AltProj is particularly notable for its significant time complexity reduction to $O(n d r_L^2 \log(1/\epsilon))$.
2. Robust Subspace Tracking (RST):
   • In contrast to the static view of RPCA, robust subspace tracking aims to adaptively follow the subspace of a time-varying data sequence. The ReProCS framework provides a robust online mechanism, balancing computational efficiency with robustness against outliers. This is crucial for applications involving long data sequences where the subspace can drift over time.
3. Robust Subspace Recovery (RSR):
   • For settings where entire vectors might be outliers, solutions like outlier pursuit extend the RPCA paradigm to facilitate RSR. This method effectively combines column-sparse matrix decomposition techniques with the goals of RPCA.

Numerical Results and Experimental Evaluation

The paper provides insightful numerical results that compare different methodologies on the CDnet 2012 dataset, a comprehensive benchmark in the domain of video analytics. It is found that, among provably robust methods, ReProCS-based approaches yield superior performance in dynamically changing environments (e.g., videos with dynamic backgrounds and intermittent object motion). The empirical evaluation further reveals the balance these methods achieve between accuracy and computational feasibility, with ReProCS exemplifying minimal time complexity while achieving optimal accuracy in complex scenarios.

Implications and Future Directions

The implications of this research extend across domains heavily reliant on dimensionality reduction techniques amidst noisy environments—sector applications spanning from video analytics to social network dynamics and beyond. Potential future directions identified include more rigorous exploration of under-sampled dynamic RPCA, along with open questions around robust subspace tracking for moving sensors.

This paper serves as an authoritative examination of robust subspace learning, offering clear guidance on algorithmic implementations and hurdles across various subspace-assessment methodologies. It forms a robust basis for expert researchers and practitioners seeking to advance the practical deployment of robust PCA and subspace learning systems in dynamically evolving contexts. The open research challenges and potential applications envisioned in this paper present rich ground for future scholarly inquiry and innovation in the field of robust data analytics.