  Converting an n-dimensional vector to a probability distribution over n
objects is a commonly used component in many machine learning tasks like
multiclass classification, multilabel classification, attention mechanisms etc.
For this, several probability mapping functions have been proposed and employed
in literature such as softmax, sum-normalization, spherical softmax, and
sparsemax, but there is very little understanding in terms how they relate with
each other. Further, none of the above formulations offer an explicit control
over the degree of sparsity. To address this, we develop a unified framework
that encompasses all these formulations as special cases. This framework
ensures simple closed-form solutions and existence of sub-gradients suitable
for learning via backpropagation. Within this framework, we propose two novel
sparse formulations, sparsegen-lin and sparsehourglass, that seek to provide a
control over the degree of desired sparsity. We further develop novel convex
loss functions that help induce the behavior of aforementioned formulations in
the multilabel classification setting, showing improved performance. We also
demonstrate empirically that the proposed formulations, when used to compute
attention weights, achieve better or comparable performance on standard seq2seq
tasks like neural machine translation and abstractive summarization.

Machine Learning;Computation and Language;Machine Learning