Robust PCA with Partial Subspace Knowledge

TL;DR

This paper introduces modified-PCP, an improved robust PCA method that leverages partial subspace knowledge to achieve accurate low-rank and sparse matrix recovery under weaker conditions, with theoretical guarantees and practical demonstrations.

Contribution

It proposes modified-PCP, a novel variation of PCP that incorporates partial subspace knowledge to enhance recovery performance and relax incoherence assumptions.

Findings

Modified-PCP requires weaker incoherence assumptions than PCP.

Theoretical correctness of modified-PCP is established.

Simulations and real data comparisons validate the approach.

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.