Recursive robust PCA or recursive sparse recovery in large but structured noise

We study the recursive robust principal components´ analysis (PCA) problem. Here, “robust” refers to robustness to both independent and correlated sparse outliers. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, S_t, in the presence of large but structured noise, L_t: the noise needs to lie in a “slowly changing” low dimensional subspace. We study a novel solution called Recursive Projected CS (ReProCS). Under mild assumptions, we show that, with high probability (w.h.p.), at all times, ReProCS can exactly recover the support set of S_t; and the reconstruction errors of both S_t and L_t are upper bounded by a time-invariant and small value.