Simpler is better: a comparative study of randomized pivoting algorithms for CUR and interpolative decompositions

Yijun Dong, Per-Gunnar Martinsson

Abstract: Matrix skeletonizations like the interpolative and CUR decompositions provide a framework for low-rank approximation in which subsets of a given matrix’s columns and/or rows are selected to form approximate spanning sets for its column and/or row space. Such decompositions that rely on “natural” bases have several advantages over traditional low-rank decompositions with orthonormal bases, including preserving properties like sparsity or non-negativity, maintaining semantic information in data, and reducing storage requirements. Matrix skeletonizations can be computed using classical deterministic algorithms such as column-pivoted QR, which work well for small-scale problems in practice, but suffer from slow execution as the dimension increases and can be vulnerable to adversarial inputs. More recently, randomized pivoting schemes have attracted much attention, as they have proven capable of accelerating practical speed, scale well with dimensionality, and sometimes also lead to better theoretical guarantees. This manuscript provides a comparative study of various randomized pivoting-based matrix skeletonization algorithms that leverage classical pivoting schemes as building blocks. We propose a general framework that encapsulates the common structure of these randomized pivoting-based algorithms and provides an a-posteriori-estimable error bound for the framework. Additionally, we propose a novel concretization of the general framework and numerically demonstrate its superior empirical efficiency.