We consider the problem of selecting the "best" subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn 2 , m 2 n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects Θ(k log k) columns according to a judiciously-chosen probability distribution that depends on information in the top-k right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let P C denote the projection matrix onto the span of those columns, and let A k denote the "best" rank-k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that, with probability at least 0.8,
Very recently, Boutsidis et al. REF introduced an algorithm for the problem of selecting exactly k columns from a matrix A to approximate A k , combining the random sampling schemes and the deterministic column pivoting strategies exploited by QR algorithms.
An Improved Approximation Algorithm for the Column Subset Selection Problem
ArXiv
ArXiv
Mathematics Computer Science