Illustration for solutions of Y = X β + z in the noiseless case (left; where z = 0) and the strong noise case (right). Each dot represents a solution (the large dot is the ground truth), where the distance to the center is the L0-norm of the solution. In the noiseless case, we only have one very sparse solution, with all other being much denser. In the strong noise case, signals are rare and weak, and we have many very sparse solutions that have comparable sparsity to that of the ground truth.