Provable Dynamic Robust PCA or Robust Subspace Tracking

Dynamic robust principal component analysis (PCA) refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lie in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, which we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. This paper provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace change, and a lower bound assumption on most outlier magnitudes. Our result is significant because: 1) it removes the strong assumptions needed by the two previous complete guarantees for ReProCS-based algorithms; 2) it shows that it is possible to achieve significantly improved outlier tolerance, compared with all existing RPCA or dynamic RPCA solutions by exploiting the above-mentioned two simple extra assumptions; and 3) it proves that simple-ReProCS is online (after initialization), fast, and, has near-optimal memory complexity.