Audio source separation is an important problem for many audio applications. Independent low-rank matrix analysis (ILRMA) is a recently proposed algorithm that employs the statistical independence between sources and the low-rankness of the time-frequency structure in each source. As reported in this paper, we have developed a new framework that enables us to introduce a spatial regularization of the demixing matrix in ILRMA. Since the conventional optimization cannot be applied to this regularized ILRMA, we derive a novel approach based on vectorwise coordinate descent, which does not require a step-size parameter and guarantees convergence. In experiments, ILRMA with beamforming-based regularization is evaluated as an application of the proposed framework.