Robust PCA with Partial Subspace Knowledge

Abstract

In recent work, robust Principal Components Analysis (PCA) has been posed as a problem of recovering a low-rank matrix $\mathbf{L}$ and a sparse matrix $\mathbf{S}$ from their sum, $\mathbf{M}:=\mathbf{L}+\mathbf{S}$ and a provably exact convex optimization solution called PCP has been proposed. This work studies the following problem. Suppose that we have partial knowledge about the column space of the low rank matrix $\mathbf{L}$. Can we use this information to improve the PCP solution, i.e. allow recovery under weaker assumptions? We propose here a simple but useful modification of the PCP idea, called modified-PCP, that allows us to use this knowledge. We derive its correctness result which shows that, when the available subspace knowledge is accurate, modified-PCP indeed requires significantly weaker incoherence assumptions than PCP. Extensive simulations are also used to illustrate this. Comparisons with PCP and other existing work are shown for a stylized real application as well. Finally, we explain how this problem naturally occurs in many applications involving time series data, i.e. in what is called the online or recursive robust PCA problem. A corollary for this case is also given.

19 pages, 9 figures, submitted to IEEE Transaction on Signal Processing

References in corpus (1)

• pROST : A Smoothed Lp-norm Robust Online Subspace Tracking Method for Realtime Background Subtraction in Video

Cited by in corpus (6)

• Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery
• Static and Dynamic Robust PCA and Matrix Completion: A Review
• Provable Dynamic Robust PCA or Robust Subspace Tracking
• Fast Robust Subspace Tracking via PCA in Sparse Data-Dependent Noise
• Manifold Denoising by Nonlinear Robust Principal Component Analysis
• Online (and Offline) Robust PCA: Novel Algorithms and Performance Guarantees