Exact recovery of low-rank plus compressed sparse matrices

Given the superposition of a low-rank matrix plus the product of a known fat compression matrix times a sparse matrix, the goal of this paper is to establish conditions under which exact recovery of the low-rank and sparse components becomes possible. This fundamental identifiability task subsumes compressed sensing and the timely low-rank plus sparse matrix recovery encountered in matrix decomposition problems. Leveraging the ability of ℓ₁- and nuclear norms to recover sparse and low-rank matrices, a convex program is formulated to estimate the unknowns. Analysis and simulations confirm that the said convex program can recover the unknowns for sufficiently low-rank and sparse enough components, along with a compression matrix possessing an isometry property.