The elastic net: Stability for sparsity methods

Abstract

In recent years, methods for sparse approximation have gained considerable attention and have been successfully applied to numerous problems in various mathematical disciplines.This work starts by illustrating applications for sparse approximation to introduce the concept of sparsity.Afterwards, the mathematical framework and basic mathematical principles are introduced.Particularly l1 minimization, which is an important tool in the sparsity context, will be introduced as well as available algorithms. This forms a profound background to approach the problem of stability in l1 minimization for ill-conditioned linear equations.It turns out that a tool arising from statistics -- the elastic net -- promises to attenuate stability problems, while preserving the benefits of l1 minimization.The connection between l1 minimization and the elastic net is discussed.Analytical properties of the elastic net are stated and corresponding algorithms are developed.Numerical troubles of l1 minimization are demonstrated for sample problems as well as the influence of the elastic net.Finally, given all necessary tools, the discussion leadsto the highlight of exact-recovery conditions for elastic-net minimization.