Interference alignment (IA) is a linear precoding strategy that can achieve optimal capacity scaling at high SNR in interference networks. However, most existing IA designs require full channel state information (CSI) at the transmitters, which would lead to significant CSI signaling overhead. There are two techniques, namely CSI quantization and CSI feedback filtering, to reduce the CSI feedback overhead. In this paper, we consider IA processing with CSI feedback filtering in MIMO cellular networks. We introduce a novel metric, namely the feedback dimension, to quantify the first order CSI feedback cost associated with the CSI feedback filtering. The CSI feedback filtering poses several important challenges in IA processing. First, there is a hidden partial CSI knowledge constraint in IA precoder design which cannot be handled using conventional IA design methodology. Furthermore, existing results on the feasibility conditions of IA cannot be applied due to the partial CSI knowledge. Finally, it is very challenging to find out how much CSI feedback is actually needed to support IA processing. We shall address the above challenges and propose a new IA feasibility condition under partial CSIT knowledge in MIMO cellular networks. Based on this, we consider the CSI feedback profile design subject to the degrees of freedom requirements, and we derive closed-form trade-off results between the CSI feedback cost and IA performance in MIMO cellular networks.