title: Imposing Uniqueness to Achieve Sparsity.
abstract: In this paper we take a novel approach to the regularization of underdetermined linear systems. Typically, a prior distribution is imposed on the unknown to hopefully force a sparse solution, which often relies on uniqueness of the regularized solution (something which is typically beyond our control) to work as desired. But here we take a direct approach, by imposing the requirement that the system takes on a unique solution. Then we seek a minimal residual for which this uniqueness requirement holds. When applied to systems with non-negativity constraints or forms of regularization for which sufficient sparsity is a requirement for uniqueness, this approach necessarily gives a sparse result. The approach is based on defining a metric of distance to uniqueness for the system, and optimizing an adjustment that drives this distance to zero. We demonstrate the performance of the approach with numerical experiments.
authors.affiliation: Department of Biomedical Engineering, Tulane University, New Orleans, LA, USA.
authors.forename: Keith
authors.initials: K
authors.lastname: Dillon
authors.affiliation: Department of Biomedical Engineering, Tulane University, New Orleans, LA, USA.
authors.forename: Yu-Ping
authors.initials: YP
authors.lastname: Wang
keywords: Convex optimization; Non-negativity; Regularization; Sparsity; Underdetermined linear systems; Uniqueness