How to solve a 1D Least Squares with $ L_1 $ Regularization?
I know gradient based method, I wonder how much faster / efficient I can get.
Related to Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring).
The solution is very similar to what I had in Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring).
The only difference in the MM is setting $ D = I $.
This means there is no reason to use the Matrix Inversion Lemma. Hence one need to prevent zeros in the values of $ {\Lambda}_{k} $.
The comparison yields:
In this case each MM iteration is 10 times slower than ADMM iteration. Still the MM convergence is much faster and at iteration 500 it is better than iteration 5000 of the ADMM.
The code is available at my StackExchange Signal Processing Q76626 GitHub Repository (Look at the SignalProcessing\Q76626 folder).
The pseudo norm $ {L}_{\frac{1}{2}} $ can be solved as described in $ {L}_{\frac{1}{2}} $ Regularization.
