Andrew E. Waters, Aswin C. Sankaranarayanan, Richard G. Baraniuk, "SpaRCS: Recovering Low-Rank and Sparse Matrices from Compressive Measurements," in Advances in Neural Information Processing Systems (NIPS), Granada, Spain, December 2011.

Abstract:  We consider the problem of recovering a matrix M that is the sum of a low-rank matrix L and a sparse matrix S from a small set of linear measurements of the form Y = A(M) = A(L+S).  This model subsumes three important classes of signal recovery problems:  compressive sensing, affine rank minimization, and robust principal component analysis.  We propose a natural optimization problem for signal recovery under this model and develop a new greedy recovery algorithm called SpaRCS.  SpaRCS inherits a number of desirable properties from the state-of-the-art CoSaMP and ADMiRA algorithms, including exponential convergence and efficient implementation.  Simulation results with video compressive sensing, hyperspectral imaging, and robust matrix completion data sets demonstrate both the accuracy and efficacy of the algorithm.

An example from the paper illustrating the efficacy of SpaRCS for video compressive sensing (CS).  (a) Several 128x128 pixel image frames from a 201 frame ground truth video.  These were sensed by a simulated single-pixel CS camera that operates independently on each image frame.  (b) The recovered low-rank component L captures the static background.  (c) The recovered sparse component S captures the people walking in the foreground. The total recovery SNR is 31.2 dB at a measurement rate of 15% of the total number of video voxels.