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In this work, we study the online robust principal components' analysis(RPCA) problem. In recent work, RPCA has been defined as a problem ofseparating a low-rank matrix (true data), $L$, and a sparse matrix (outliers),$S$, from their sum, $M:=L + S$. A more general version of this problem is torecover $L$ and $S$ from $M:=L + S + W$ where $W$ is the matrix of unstructuredsmall noise/corruptions. An important application where this problem occurs isin video analytics in trying to separate sparse foregrounds (e.g., movingobjects) from slowly changing backgrounds. While there has been a large amountof recent work on solutions and guarantees for the batch RPCA problem, theonline problem is largely open."Online" RPCA is the problem of doing the aboveon-the-fly with the extra assumptions that the initial subspace is accuratelyknown and that the subspace from which $l_t$ is generated changes slowly overtime. We develop and study a novel "online" RPCA algorithm based on therecently introduced Recursive Projected Compressive Sensing (ReProCS)framework. Our algorithm improves upon the original ReProCS algorithm and italso returns even more accurate offline estimates. The key contribution of thiswork is a correctness result (complete performance guarantee) for thisalgorithm under reasonably mild assumptions. By using extra assumptions --accurate initial subspace knowledge, slow subspace change, and clusteredeigenvalues -- we are able to remove one important limitation of batch RPCAresults and two key limitations of a recent result for ReProCS for online RPCA.To our knowledge, this work is among the first few correctness results foronline RPCA. Most earlier results were only partial results, i.e., theyrequired an assumption on intermediate algorithm estimates.

Online (and Offline) Robust PCA: Novel Algorithms and Performance Guarantees