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

Abstract

SummaryThe Lawson‐Hanson with Deviation Maximization (LHDM) method is a block algorithm for the solution of NonNegative Least Squares (NNLS) problems. In this work we devise an improved version of LHDM and we show that it terminates in a finite number of steps, unlike the previous version, originally developed for a special class of matrices. Moreover, we are concerned with finding sparse solutions of underdetermined linear systems by means of NNLS. An extensive campaign of experiments is performed in order to evaluate the performance gain with respect to the standard Lawson‐Hanson algorithm. We also show the ability of LHDM to retrieve sparse solutions, comparing it against several ‐minimization solvers in terms of solution quality and time‐to‐solution on a large set of dense instances.

Low cardinality positive interior cubature on NURBS-shaped domains

“…Halton sequences, in the domain, where we used our very recent implementation of the indicator function of NURBS-shaped domains via a covering of the boundary by "monotone boxes" and an economy use of the crossing number [24]. On the other hand, the moment-matching NNLS problem is coped via the classical Lawson-Hanson active-set algorithm, which naturally seeks a sparse solution, or one of its recent variants [15,9,11,20].…”

“…We point out that there are several alternative implementations available in MATLAB, as the built-in function lsqnonneg or the open-source routine of the package NNLSlab in [20]. On the other hand, in the recent papers [9,11] an acceleration of the Lawson-Hanson algorithm has been discussed in the framework of compression of discrete probability measures and regression, based on the concept of "Deviation Maximization" instead of standard column pivoting for the underlying QR factorizations [10]. Such a method, called LHDM, shows remarkable speed-ups and in perspective could be applied also in the cubature framework.…”