James Folberth

Stephen Becker, James Folberth, Laura Grigori

The unpivoted and pivoted Householder QR factorizations are ubiquitous in numerical linear algebra. A difficulty with pivoted Householder QR is the communication bottleneck introduced by pivoting. In this paper we propose using random orthogonal systems to quickly mix together the columns of a matrix before computing an unpivoted QR factorization. This method computes a URV factorization which forgoes expensive pivoted QR steps in exchange for mixing in advance, followed by a cheaper, unpivoted QR factorization. The mixing step typically reduces the variability of the column norms, and in certain experiments, allows us to compute an accurate factorization where a plain, unpivoted QR performs poorly. We experiment with linear least-squares, rank-revealing factorizations, and the QLP approximation, and conclude that our randomized URV factorization behaves comparably to a similar randomized rank-revealing URV factorization, but at a fraction of the computational cost. Our experiments provide evidence that our proposed factorization might be rank-revealing with high probability.

James Folberth, Stephen Becker

Inspired by recent work on safe feature elimination for $1$-norm regularized least-squares, we develop strategies to eliminate features from convex optimization problems with non-negativity constraints. Our strategy is safe in the sense that it will only remove features/coordinates from the problem when they are guaranteed to be zero at a solution. To perform feature elimination we use an accurate, but not optimal, primal-dual feasible pair, making our methods robust and able to be used on ill-conditioned problems. We supplement our feature elimination problem with a method to construct an accurate dual feasible point from an accurate primal feasible point; this allows us to use a first-order method to find an accurate primal feasible point, then use that point to construct an accurate dual feasible point and perform feature elimination. Under reasonable conditions, our feature elimination strategy will eventually eliminate all zero features from the problem. As an application of our methods we show how safe feature elimination can be used to robustly certify the uniqueness of non-negative least-squares (NNLS) problems. We give numerical examples on a well-conditioned synthetic NNLS problem and a on set of 40000 extremely ill-conditioned NNLS problems arising in a microscopy application.