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In this paper, we study the problem of decomposing a superposition of alow-rank matrix and a sparse matrix when a relatively few linear measurementsare available. This problem arises in many data processing tasks such asaligning multiple images or rectifying regular texture, where the goal is torecover a low-rank matrix with a large fraction of corrupted entries in thepresence of nonlinear domain transformation. We consider a natural convexheuristic to this problem which is a variant to the recently proposed PrincipalComponent Pursuit. We prove that under suitable conditions, this convex programguarantees to recover the correct low-rank and sparse components despitereduced measurements. Our analysis covers both random and deterministicmeasurement models.

Principal Component Pursuit with Reduced Linear Measurements