  We study the generalized degrees of freedom (gDoF) of the block-fading
noncoherent diamond (parallel relay) wireless network with asymmetric
distributions of link strengths, and a coherence time of T symbol duration. We
first derive an outer bound for this channel and then derive the optimal
signaling structure for this outer bound. Using the optimal signaling structure
we solve the outer bound optimization problem in terms of its gDoF. Using
insights from our outer bound signaling solution, we devise an achievability
strategy based on a novel scheme that we call train-scale quantize-map-forward
(TS-QMF). This uses training in the links from the source to the relays,
scaling and quantizing at the relays combined with nontraining-based schemes.
We show the optimality of this scheme with respect to the outer bound in terms
of the gDoF. In noncoherent point-to-point multiple-input-multiple-output
(MIMO) channels, where the fading channel is unknown to transmitter and
receiver, an important tradeoff between communication and channel learning was
revealed by Zheng and Tse, by demonstrating that not all the available antennas
might be used, as it is suboptimal to learn all their channel parameters. Our
results in this paper for the diamond network demonstrates that in certain
regimes the optimal scheme uses a subnetwork, demonstrating a tradeoff between
channel learning and communications. In some regimes, it is gDoF optimal to do
relay selection, i.e, use a part of the network. In the other regimes, even
when it is essential to use the entire network, it is suboptimal to learn the
channel states for all the links in the network, i.e, traditional
training-based schemes are suboptimal in these regimes.
