  We characterize the degrees of freedom (DoF) of multi-way relay MIMO
interference networks. In particular, we consider a wireless network consisting
of 4 user nodes, each with M antennas, and one N-antenna relay node. In this
network, each user node sends one independent message to each of the other user
nodes, and there are no direct links between any two user nodes, i.e., all
communication must pass through the relay node. For this network, we show that
the symmetric DoF value per message is given by max(min(M/3,N/7),min(2M/7,N/6))
normalized by space dimensions, i.e., piecewise linear depending on M and N
alternatively. While the information theoretic DoF upper bound is established
for every M and N, the achievability relying on linear signal subspace
alignment is established in the spatially-normalized sense in general. In
addition, by deactivating 4 messages to form a two-way relay MIMO X channel, we
also present the DoF result in the similar piecewise linear type. The central
new insight to emerge from this work is the notion of inter-user signal
subspace alignment incorporating the idea of network coding, which is the key
to achieve the optimal DoF for multi-way relay interference networks. Moreover,
this work also settles the feasibility of linear interference alignment that
extends the feasibility framework from one-way to multi-way relay interference
networks.
