  This paper continues the Wu-Shamai-Verdu program [3] on characterizing the
degrees of freedom (DoF) of interference channels (ICs) through Renyi
information dimension. Specifically, we find a single-letter formula for the
DoF of vector ICs, encompassing multiple-input multiple-output (MIMO) ICs,
time- and/or frequency-selective ICs, and combinations thereof, as well as
scalar ICs as considered in [3]. The DoF-formula we obtain lower-bounds the DoF
of all channels--with respect to the choice of the channel matrix--and
upper-bounds the DoF of almost all channels. It applies to a large class of
noise distributions, and its proof is based on an extension of a result by
Guionnet and Shlyakthenko [3] to the vector case in combination with the Ruzsa
triangle inequality for differential entropy introduced by Kontoyiannis and
Madiman [4]. As in scalar ICs, achieving full DoF requires the use of singular
input distributions. Strikingly, in the vector case it suffices to enforce
singularity on the joint distribution of each individual transmit vector. This
can be realized through signaling in subspaces of the ambient signal space,
which is in accordance with the idea of interference alignment, and, most
importantly, allows the scalar entries of the transmit vectors to have
non-singular distributions. The DoF-formula for vector ICs we obtain enables a
unified treatment of "classical" interference alignment a la Cadambe and Jafar
[5], and Maddah-Ali et al. [6], and the number-theoretic schemes proposed in
[7], [8]. Moreover, it allows to calculate the DoF achieved by new signaling
schemes for vector ICs. We furthermore recover the result by Cadambe and Jafar
on the non-separability of parallel ICs [9] and we show that almost all
parallel ICs are separable in terms of DoF. Finally, our results apply to
complex vector ICs, thereby extending the main findings of [2] to the complex
case.
