A Dictionary Based Generalization of Robust PCA

ABSTRACT We analyze the decomposition of a data matrix, assumed to be a superposition of a low-rank component and a component which is sparse in a known dictionary, using a convex demixing method.We provide a unified analysis, encompassing both undercomplete and overcomplete dictionary cases, and show that the constituent components can be successfully recovered undersome relatively mild assumptions up to a certain global sparsity level. Further, we corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity for various dictionary sizes. Index Terms-- Low-rank, dictionary sparse, Robust PCA. 1. INTRODUCTION Exploiting the inherent structure of data for the recovery of relevant information is at the heart of data analysis. R. A wide range of problems can be expressed in the form described above. Perhaps the most celebrated of these is principal componentanalysis (PCA) [1], which can be viewed as a special case of eq.(1), with the matrix X, the problem reduces to that of sparse recovery [2-4]; See [5] and references therein for an overview of related works.