How the CVX implement optimization with L1-Norm regularization?

Which algorithm CVX used to do it? I implement soft thresholding to do L1-Norm regularization. But I found the objective value is not as good as the result of CVX.

All of CVX’s solvers use an interior-point method, which is a far more general-purpose method. It tends to be more accurate, as you have observed, but it is usually slower. Most interior-point methods are not going to provide good performance for large-scale sparse recovery problems, although some recent work by Jacek Gondzio suggests that a carefully designed interior-point method could be quite competitive.

I used cvx and OMP at the same time I think cvx is not good as OMP. Am I wrong ?