Parsimonious Least Norm Approximation

Abstract

A theoretically justifiable fast finite successive linear approximation algorithm is proposed for obtaining a parsimonious solution to a corrupted linear system Ax=b+p, where the corruption p is due to noise or error in measurement. The proposed linear-programming-based algorithm finds a solution x by parametrically minimizing the number of nonzero elements in x and error ||Ax-b-p||1. Numerical tests on a signal-processing-based example indicate that the proposed method is comparable to a method that parametrically minimizes the 1-norm of the solution x and the error ||Ax-b-p||1, and that both methods are superior, by orders of magnitude, to solutions obtained by least squares as well by combinatorially choosing an optimal solution with a specific number of nonzero elements.