Fig. 2
Non-smooth penalties shift the objective function minimum to zero. Shift of the objective function minimum towards zero for increasing penalty strength. The horizontal positions of the diamond tips mark the global minimum of the regularized objective function with the penalty strength denoted inside. The black curves represent an unpenalized objective function \(\chi ^{2}_{\text {ML}}\). Dark-gray curves depict the sum of \(\chi ^{2}_{\text {ML}}\) and a penalty with strength λ=1. The filled diamonds in zero represent penalty strengths λZ which cause the minimum to be exactly in zero. The objective function penalized with λZ is drawn in light grey. Finite λZ=2 is sufficient to shift the minimum to zero. While the convex absolute-value penalty only admits one minimum, the non-convex ℓq penalty can lead to multiple local minima as depicted for λ=1 on the right hand panel. For most optimizations described in this manuscript, an ℓ0.8 penalty was used. In this figure, the case of q=0.5 is depicted to show the multiple minima more clearly. They arise, however, for any 0<q<1