Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise

Abstract

Abstract-This work studies the recursive robust principal components analysis (PCA) problem. If the outlier is the signalof-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low dimensional subspace that is either fixed or changes "slowly enough." A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above problem, in recent work, we introduced a novel solution called Recursive Projected CS (ReProCS). In this work we develop a simple modification of the original ReProCS idea and analyze it. This modification assumes knowledge of a subspace change model on the Lt's. Under mild assumptions and a denseness assumption on the unestimated part of the subspace of Lt at various times, we show that, with high probability (w.h.p.), the proposed approach can exactly recover the support set of St at all times; and the reconstruction errors of both St and Lt are upper bounded by a time-invariant and small value. In simulation experiments, we observe that the last assumption holds as long as there is some support change of St every few frames.

Recursive projected sparse matrix recovery (ReProSMR) with application in real-time video layer separation

“…However, foreground objects typically occupy contiguous regions and move in a correlated fashion resulting S being sparse as well as low rank. As we observed in [9,10,11], PCP would fail if the support size of S is not small and S has low rank. To address this problem, we introduced an online approach in [9] which we later called Recursive Projected Compressive Sensing (ReProCS) in [10,11].…”

“…This work designed and evaluated ReProSMR, which is a modification of our earlier work [9,10,11]. We showed that ReProSMR has excellent performance for a real-time video layer separation problem and the performance is better than many of the state-of-the-art algorithms from recent work.…”

A Unified View of Nonconvex Heuristic Approach for Low-Rank and Sparse Structure Learning

“…Even for offline applications, a recursive solution is typically faster than a batch one. In [43], we studied a simple modification of the original ReProCS idea and obtained a performance guarantee for it. In recent work [22,39,40,43], we introduced a novel solution approach, called Recursive Projected Compressive Sensing (ReProCS), which recursively recovered S t and L t at each time t. Moreover, as we showed later in [31,32], it can also provably handle correlated support changes significantly better than batch approaches because it uses extra assumptions (accurate initial subspace knowledge and slow subspace change).…”

A robust online subspace estimation and tracking algorithm

“…Recently, several algorithms have been proposed to address the online subspace estimation problem from incomplete observations [14][15][16][17][18][19][20]. The GROUSE algorithm [14] uses rankone updates of the estimated subspace on the Grassmannian manifold.…”

“…More recently, the GRASTA algorithm [16], or robust GROUSE, also uses updates on the Grassmannian manifold using a robust 1 -norm cost to recover from outliers in the observations. Other robust online PCA techniques include recursive projection in ReProCS [17], bilinear decomposition [18], and adaptive projected subgradient STAPSM [19]. These algorithms either do not handle missing data or require relatively accurate initial estimates of the target subspace.…”