Lower bound theory of nonzero entries in solutions of l ₂-l [sub p] minimization

Abstract

Recently, variable selection and sparse reconstruction are solved by finding an optimal solution of a minimization model, where the objective function is the sum of a data-fitting term in l₂ norm and a regularization term in l [sub p] norm (0<p<1). Since it is a nonconvex model, most algorithms for solving the problem can provide only an approximate local optimal solution, where nonzero entries in the solution cannot be identified theoretically. In this paper, we establish lower bounds for the absolute value of nonzero entries in every local optimal solution of the model, which can be used to indentify zero entries precisely in any numerical solution. Therefore, we have developed a lower bound theorem to classify zero and nonzero entries in every local solution. These lower bounds clearly show the relationship between the sparsity of the solution and the choice of the regularization parameter and norm so that our theorem can be used for selecting desired model parameters and norms. Furthermore, we also develop error bounds for verifying the accuracy of numerical solutions of the l₂-l [sub p] minimization model. To demonstrate applications of our theory, we propose a hybrid orthogonal matching pursuit-smoothing gradient (OMP-SG) method for solving the nonconvex, non-Lipschitz continuous l₂-l [sub p] minimization problem. Computational results show the effectiveness of the lower bounds for identifying nonzero entries in numerical solutions and the OMP-SG method for finding a high quality numerical solution.