We study distributed rank selection in a multi- cell wideband cellular network with multiple antennas per node. The rank for a particular mobile is the number of independent data streams across all sub-carriers and antennas. Rank selection then involves both spatial multiplexing and sub-carrier allocations. We propose assigning frequency-space signatures across the data streams to unify the rank assignment over those two dimensions. A sum rate maximization problem is formulated for allocating the signatures and is solved using distributed bi-directional training. Starting with full-rank transmission, the precoders are iteratively updated, and the optimized rank for a mobile is the number of signatures assigned significant power after convergence. For systems with a large number of antennas and sub-carriers, it may take many iterations to converge. To reduce the training period, we propose using deep learning for estimating the ranks across mobiles before the precoder optimization phase. With a trained Deep Neural Network (DNN), each mobile estimates its own transmission rank directly from its local channel information and locations of neighboring mobiles. Simulation results show that the DNNs can significantly reduce the training period for precoder adaptation while achieving performance close to that with jointly optimized precoders and ranks. This is illustrated with different system parameters and mobile distributions.