Occlusion, real disguise and illumination are still the common difficulties encountered in face recognition. The sparse representation based classifier (SRC) has shown a great potential in handling pixel-level sparse noise, while the nuclear norm based matrix regression (NMR) model has been demonstrated to be powerful for dealing with the image-wise structural noise. Both methods, however, might be not very effective for handling the mixed noise: the structural noise plus the sparse noise. In this paper, we present two nuclear-L₁ norm joint matrix regression (NL₁R) models for face recognition with mixed noise, which are derived by using MAP (maximum a posteriori probability estimation). The first model considers the mixed noise as a whole, while the second model assumes the mixed noise is an additive combination of two independent componenral nts: sparse noise and structuoise. The proposed models can be solved by the alternating direction method of multipliers (ADMM). We validate the effectiveness of the proposed models through a series of experiments on face reconstruction and recognition.