Fig. 2.

For both linear regression (A1, A2) and regression with outliers (B1, B2) from Scenario I, MPRidge attains nearly the same out-of-sample risk as the explicitly ridge-regularized counterpart: in A1 and B1, the α- and λ-axes are aligned based on in-sample risk, so that R_ens, in (green dot) aligns perfectly with R_{ridge, in} (blue triangle). This results in the out-of-sample risk R_ens, out (purple dot) also aligning approximately with R_{ridge, out} (red triangle). Additionally, MPRidge largely recovers the corresponding regularization path, as shown in A2 and B2. This suggests that our MPRidge method elicits ridge-like regularizing effects implicitly.