Provable Dynamic Robust PCA or Robust Subspace Tracking

Abstract

Dynamic robust PCA refers to the dynamic (timevarying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies 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, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work 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 (i) it removes the strong assumptions needed by the two previous complete guarantees for ReProCS-based algorithms; (ii) 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 two simple extra assumptions; and (iii) it proves that simple-ReProCS is online (after initialization), fast, and, has nearoptimal memory complexity.

Provable Low Rank Phase Retrieval

“…Other subspace tracking (ST) problems that have been extensively studied include dynamic compressive sensing [58] (a special case of ST where the subspace is defined by the span of a subset of r vectors from a known dictionary matrix), dynamic robust PCA (or robust ST), see [55], [56] and references therein, streaming PCA with missing data [57], [59], and ST with missing data [60]- [64]. In terms of works with complete provable guarantees, there is the nearly optimal robust subspace tracking via recursive projected compressive sensing approach [55], [56], [64] and its precursors; recent papers on streaming PCA with missing data [57], [59], and older work on dynamic compressive sensing (CS) [58]. For robust ST, the problem setting itself implies m = n/2.…”

Provable Subspace Tracking From Missing Data and Matrix Completion

“…Compressive Sensing (ReProCS) [3], [18]- [21]. Robust STmiss has not received much attention in the literature.…”

Phaseless Subspace Tracking

“…A less general model, but one that allows for a reduction in the number of unknowns, is to assume that the true data subspace is piecewise constant with time. This model has been extensively used in robust subspace tracking literature [8,9,10] where it in fact helps ensure identifiability of the subspaces (in that problem, only one n length measurement vector is available at each time t).…”

“…Assumptions. We quantify "slow subspace change" using the model from [9]. In [9] and previous work, this has been successfully used to improve outlier tolerance of dynamic robust PCA as compared to its static counterpart.…”

“…We quantify "slow subspace change" using the model from [9]. In [9] and previous work, this has been successfully used to improve outlier tolerance of dynamic robust PCA as compared to its static counterpart. "Slow subspace change" [9] means that U t = U tj := U j ∀ t ∈ [t j : t j+1 ) (piecewise constant subspaces) and the following hold: (a) SE(U j−1 , U j ) ≤ ∆ with ∆ small, (b) at each change time, only one direction changes, and (c) min j (t j+1 − t j ) is lower bounded.…”