In a paper, I read that this is one problem with the L1 norm for Compressed sensing (the L1 norm function is not smooth). I am curious that why is that a problem? Can anyone explain why is it so important to be smooth? I guess it is about constrained optimization. Thank you.
Yes, being non-differentiable (at 0) makes it hard to optimize. Most optimization methods require the derivative. Although the L1 norm is convex, being non-differentiable means that special optimization methods have had to be developed to try to solve problems that use the L1 norm, such as in compressed sensing.