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In K-user multiple-input multiple-output (MIMO) interference channels, it was shown that interference alignment (IA) achieves a full spatial multiplexing gain when perfect channel state information (CSI) is available at each transmitter in the network. When the CSI is fed back from receivers using the limited number of feedback bits, a significant performance loss is inevitable in the IA due to quantized channel knowledge. In this paper, we propose a new channel quantization strategy to optimize the performance of the IA with limited feedback. In our proposed scheme, we introduce an additional receive filter to minimize the chordal distance which accounts for the quantization error on Grassmann manifold. Besides, we analyze a reduction in terms of the chordal distance in our scheme compared to conventional methods. Simulation results verify that the proposed scheme provides substantially better performance than the conventional method as the number of feedback bits is increased. We show that our scheme exhibits 30% and 40% sum rate gains compared to the conventional scheme when the numbers of the feedback bits are 10 and 15, respectively, with two antennas per node.