Stable Recovery of Sparse Signals Via Regularized Minimization

Suppose we want to recover a vector x₀isinR^m from incomplete and contaminated measurements y=Ax₀+e, where A is a p by m matrix, p<m and e is an error term, parepar₂=epsiv. In previous work, Donoho et al. (1998), Candes et al. (Commun. Pure Appl.) have shown that if the signal x₀ is sufficiently sparse, it can be recovered stably by lscr₁-minimization. More precisely, the vector x^# with minimal lscr₁ norm, constrained by parAx^#-ypar₂lesepsiv, satisfies parx^#-x₀par₂lesCepsiv. In the present correspondence it is shown that one can also choose a regularization approach, in which one minimizes the functional parAx-ypar₂ ²+muparxpar₁, without additional constraints; its minimizer x* is likewise a good estimate for x₀, with a bound on parx*-x₀par₂ proportional to epsiv.