A new piecewise quadratic approximation approach for L0 norm minimization problem

Abstract

In this paper, we consider the problem of finding sparse solutions for underdetermined systems of linear equations, which can be formulated as a class of $L_{0}$ norm minimization problem. By using the least absolute residual approximation, we propose a new piecewise quadratic function to approximate the $L_{0}$ norm. Then, we develop a piecewise quadratic approximation (PQA) model where the objective function is given by the summation of a smooth non-convex component and a non-smooth convex component. To solve the (PQA) model, we present an algorithm based on the idea of the iterative thresholding algorithm and derive the convergence and the convergence rate. Finally, we carry out a series of numerical experiments to demonstrate the performance of the proposed algorithm for (PQA). We also conduct a phase diagram analysis to further show the superiority of (PQA) over $L_{1}$ and $L_{1/2}$ regularizations.