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

Abstract

<p>In this work, we study the problem of recursively recovering a time sequence of sparse vectors, S_t, from measurements M_t := S_t + L_t that are corrupted by structured noise L_t which is dense and can have large magnitude. The structure that we require is that L_t should lie in a low dimensional subspace that is either fixed or changes ``slowly enough"; and the eigenvalues of its covariance matrix are ``clustered". We do not assume anything about the sequence of sparse vectors, except a bound on their support size. Their support sets and their nonzero element values may be either independent or correlated over time (usually in many applications they are correlated). A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (L_t) from moving foreground objects (S_t) on-the-fly. To solve the above problem, we introduce a novel solution called Recursive Projected Compressive Sensing (ReProCS). Under mild assumption, we show that ReProCS can exactly recover the support set of S_t at all times; and the reconstruction errors of both S_t and L_t are upper bounded by a time-invariant and small value at all times. ReProCS is designed under the assumption that the subspace in which the most recent several L_t's lie can only grow over time. Therefore, it needs to assume a bound on the total number of subspace changes, $J$. To address this limitation, we introduce a novel subspace estimation scheme called cluster-PCA and we refer to the resulting algorithm as ReProCS with cluster-PCA (ReProCS-cPCA). ReProCS-cPCA does not need a bound on J as long as the delay between subspace change times increases in proportion to <em>log J</em>. An extra assumption that is needed though is that the eigenvalues of the covariance matrix of L_t are sufficiently clustered. As a by-product, at certain times, the basis vectors for the subspace in which the most recent several L_t's lies is also recovered.</p>