Lower Bound Theory of Nonzero Entries in Solutions of ℓ2-ℓp 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 ℓp norm (0 < p < 1). Since it is a non-convex model, most algorithms for solving the problem can only provide 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 eliminate zero entries precisely in any numerical solution. Therefore, we have developed a lower bound theorem to classify zero and nonzero entries in its 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 accuracy of numerical solutions of the ℓ2-ℓp 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-ℓp 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.