  While interference alignment schemes have been employed to realize the full
multiplexing gain of $K$-user interference channels, the analyses performed so
far have predominantly focused on the case when global channel knowledge is
available at each node of the network. This paper considers the problem where
each receiver knows its channels from all the transmitters and feeds back this
information using a limited number of bits to all other terminals. In
particular, channel quantization over the composite Grassmann manifold is
proposed and analyzed. It is shown, for $K$-user multiple-input,
multiple-output (MIMO) interference channels, that when the transmitters use an
interference alignment strategy as if the quantized channel estimates obtained
via this limited feedback are perfect, the full sum degrees of freedom of the
interference channel can be achieved as long as the feedback bit rate scales
sufficiently fast with the signal-to-noise ratio. Moreover, this is only one
extreme point of a continuous tradeoff between achievable degrees of freedom
region and user feedback rate scalings which are allowed to be non-identical.
It is seen that a slower scaling of feedback rate for any one user leads to
commensurately fewer degrees of freedom for that user alone.
