ReProCS: A Missing Link between Recursive Robust PCA and Recursive Sparse Recovery in Large but Correlated Noise

This work studies the recursive robust principal components analysis (PCA) problem. Here, robust refers to robustness to both independent and correlated sparse outliers, although we focus on the latter. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background from moving foreground objects on-the-fly. The background sequence is well modeled as lying in a low dimensional subspace, that can gradually change over time, while the moving foreground objects constitute the correlated sparse outliers. In this and many other applications, the foreground is an outlier for PCA but is actually the signal of interest for the application; where as the background is the corruption or noise. Thus our problem can also be interpreted as one of recursively recovering a time sequence of sparse signals in the presence of large but spatially correlated noise. This work has two key contributions. First, we provide a new way of looking at this problem and show how a key part of our solution strategy involves solving a noisy compressive sensing (CS) problem. Second, we show how we can utilize the correlation of the outliers to our advantage in order to even deal with very large support sized outliers. The main idea is as follows. The correlation model applied to the previous support estimate helps predict the current support. This prediction serves as partial support knowledge for solving the modified-CS problem instead of CS. The support estimate of the modified-CS reconstruction is, in turn, used to update the correlation model parameters using a Kalman filter (or any adaptive filter). We call the resulting approach support-predicted modified-CS.