  We investigate the total degrees of freedom (DoF) of the K-user
rank-deficient interference channel with feedback. For the two-user case, we
characterize the total DoF by developing an achievable scheme and deriving a
matching upper bound. For the three-user case, we develop a new achievable
scheme which employs interference alignment to efficiently utilize the
dimension of the received signal space. In addition, we derive an upper bound
for the general K-user case and show the tightness of the bound when the number
of antennas at each node is sufficiently large. As a consequence of these
results, we show that feedback can increase the DoF when the number of antennas
at each node is large enough as compared to the ranks of channel matrices. This
finding is in contrast to the full-rank interference channel where feedback
provides no DoF gain. The gain comes from using feedback to provide alternative
signal paths, thereby effectively increasing the ranks of desired channel
matrices.
