The original interference alignment (IA) scheme provides poor sum-rate performance compared to simple orthogonal access schemes such as time-division multiple access (TDMA) in low-to-medium SNR under total power constraint. In this paper, we address this problem by proposing a method of regularizing the IA scheme with a criterion of minimizing the weighted sum of the mean square error (WMSE) function. To perform the regularization process efficiently, the weight terms in the WMSE metric should be computed from the optimal zero-forcing (ZF) schemes. Thus, we first prove the optimality of the enhanced IA algorithm introduced in our previous work in the ZF sense. From simulation results, it is shown that the proposed scheme outperforms the TDMA in overall SNR regime. We can further improve the performance by repeating the proposed regularization process iteratively. Moreover, we propose a modified design that provides robustness in the presence of channel uncertainty.