Deviation maximization for rank-revealing QR factorizations

Abstract

In this paper, we introduce a new column selection strategy, named here “Deviation Maximization”, and apply it to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK’s routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present experimental results on a wide data set of numerically singular matrices, which has become a reference in the recent literature.

The Lawson‐Hanson algorithm with deviation maximization: Finite convergence and sparse recovery

“…If u 2 is the vector of column norms of C, then the deviation maximization can select numerically linearly independent columns. 20 Here we introduce u 1 as the steepest descend direction at the current solution point in order to take into account first order information to fit the Lawson-Hanson algorithm.…”

“…The set of candidate $$$ I $$$ is made of those column indices such that the corresponding entries of $$$ {\mathbf{u}}_1,{\mathbf{u}}_2 $$$ are "large enough" with respect to thresholds $$$ {\tau}_1,{\tau}_2 $$$, respectively. If $$$ {\mathbf{u}}_2 $$$ is the vector of column norms of $$$ C $$$, then the deviation maximization can select numerically linearly independent columns 20 . Here we introduce $$$ {\mathbf{u}}_1 $$$ as the steepest descend direction at the current solution point in order to take into account first order information to fit the Lawson‐Hanson algorithm.…”

“…In this work, we address the NNLS problem in which the objective function matrix is dense and we present a modified Lawson‐Hanson algorithm in which multiple column selection is performed in order to exploit BLAS–3 operations for efficiency. The column selection is based on the “deviation maximization” pivoting, 20 which identifies a block of sufficiently linearly independent columns for QR computations. Here, the column selection takes into account first order information to solve the nonnegative least squares problem.…”

“…. However, it is known 20 that even if the matrix A (s) P is well conditioned, there could be a potentially dramatic increase in the condition number of A (s+1) P when using the deviation maximization strategy, as well as for the standard Lawson-Hanson algorithm. A famous example is the Kahan matrix, however it is a very unlikely occurrence in practice.…”

“…Equation (10) implies in particular 𝛿 ≤ 𝜏 2 . However, the values of 𝛿 and 𝜏 2 can be set independently with minor modifications 20 without compromising the nonsingularity of R (s+1) . Therefore, from now on we will not require (10) In this work, we add the loop in steps 14-20, and it plays a central role in proving the following result about finite termination.…”