High dimensional decomposition of coherent/structured matrices via sequential column/row sampling

Abstract

This paper focuses on the low rank plus sparse matrix decomposition problem in big data settings. Conventional algorithms solve high-dimensional optimization problems that scale with the data dimension, which limits their scalability. In addition, existing randomized approaches mostly rely on blind random sampling. In this paper, the drawbacks of random sampling from coherent/structured data matrices are analyzed showing that random sampling cannot provide efficient descriptive sketches of coherent data. In addition, a column/row subspace pursuit algorithm which recovers the low rank component via a small set of informative columns/rows of the data is proposed. The obtained column and row spaces are updated in each iteration to converge to the column and row spaces of the low rank matrix. The informative columns are located using the information embedded in the row space while the informative rows are identified using the information embedded in the column space.