A Dictionary-Based Generalization of Robust PCA Part I: Study of Theoretical Properties

We consider the decomposition of a data matrix assumed to be a superposition of a low-rank matrix and a component which is sparse in a known dictionary, using a convex demixing method. We consider two sparsity structures for the sparse factor of the dictionary sparse component, namely entry-wise and column-wise sparsity, and provide a unified analysis, encompassing both undercomplete and the overcomplete dictionary cases, to show that the constituent matrices can be successfully recovered under some relatively mild conditions on incoherence, sparsity, and rank. We corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity, in comparison to related techniques. Investigation of a specific application in hyperspectral imaging is included in an accompanying paper.