Non-smooth equations based method for ℓ1ℓ1-norm problems with applications to compressed sensing

In this paper, we propose, analyze, and test a new method for solving ℓ1ℓ1-norm regularization problems arising from the spare solution recovery in compressed sensing, basis-pursuit problems, and linear inverse problems. The method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and a ℓ1ℓ1-norm regularization term. The problem is firstly formulated for a convex quadratic program problem, and then for an equivalent non-smooth equation. At each iteration, a spectral gradient method is applied to the resulting problem without requiring Jacobian matrix information. Convergence of the proposed method follows directly from the results which already exist. The algorithm is easily performed, where only the matrix–vector inner product is required at each and every step. Numerical experiments to decode a sparse signal arising in compressed sensing and image deconvolution are performed. The numerical results illustrate that the proposed method is practical and promising.