  The Softmax function on top of a final linear layer is the de facto method to
output probability distributions in neural networks. In many applications such
as language models or text generation, this model has to produce distributions
over large output vocabularies. Recently, this has been shown to have limited
representational capacity due to its connection with the rank bottleneck in
matrix factorization. However, little is known about the limitations of
Linear-Softmax for quantities of practical interest such as cross entropy or
mode estimation, a direction that we explore here. As an efficient and
effective solution to alleviate this issue, we propose to learn parametric
monotonic functions on top of the logits. We theoretically investigate the rank
increasing capabilities of such monotonic functions. Empirically, our method
improves in two different quality metrics over the traditional Linear-Softmax
layer in synthetic and real language model experiments, adding little time or
memory overhead, while being comparable to the more computationally expensive
mixture of Softmaxes.
