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Interference alignment (IA) is an effective scheme for counteracting multi-user interference in wireless networks. Unfortunately, IA is sensitive to channel-state-information (CSI) imperfections. Achieving perfect CSI knowledge at a central node in large scale antenna wireless networks implies a huge feedback which is prohibitive. In this paper, we assume a distributed multicellular scenario, where there is no central node that knows global CSI and optimizing IA's precoder and combiner matrices is done by exchanging the local channel information between users and base stations (BSs) in several iterations. By using low-rank matrix approximation theory, we propose an efficient method to iteratively optimize precoder and combiner matrices for distributed IA. In each iteration, only a part of the CSI is fed back to BSs. More precisely, based on the latest available CSI and certain performance criteria, a few columns of the effective channel are sent back to the transmitters to approximate the interference covariance matrix which is then used to update the precoder matrices. We also propose a new method for quantizing the channel information matrix non-uniformly, which improves upon the conventional channel feedback quantization techniques. We evaluate the proposed methods by simulating a cellular network with various number of BS antennas and different feedback channel capacities. Simulation results show that our methods outperform both the conventional and improved channel feedback quantization algorithms significantly. [ABSTRACT FROM AUTHOR]