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The CUR decomposition is a technique for low-rank approximation that selectssmall subsets of the columns and rows of a given matrix to use as bases for itscolumn and rowspaces. It has recently attracted much interest, as it hasseveral advantages over traditional low rank decompositions based onorthonormal bases. These include the preservation of properties such assparsity or non-negativity, the ability to interpret data, and reduced storagerequirements. The problem of finding the skeleton sets that minimize the normof the residual error is known to be NP-hard, but classical pivoting schemessuch as column pivoted QR work tend to work well in practice. When combinedwith randomized dimension reduction techniques, classical pivoting basedmethods become particularly effective, and have proven capable of very rapidlycomputing approximate CUR decompositions of large, potentially sparse,matrices. Another class of popular algorithms for computing CUR de-compositionsare based on drawing the columns and rows randomly from the full index sets,using specialized probability distributions based on leverage scores. Suchsampling based techniques are particularly appealing for very large scaleproblems, and are well supported by theoretical performance guarantees. Thismanuscript provides a comparative study of the various randomized algorithmsfor computing CUR decompositions that have recently been proposed.Additionally, it proposes some modifications and simplifications to theexisting algorithms that leads to faster execution times.

Simpler is better: A comparative study of randomized algorithms for computing the CUR decomposition