  Many applications of generative models rely on the marginalization of their
high-dimensional output probability distributions. Normalization functions that
yield sparse probability distributions can make exact marginalization more
computationally tractable. However, sparse normalization functions usually
require alternative loss functions for training since the log-likelihood is
undefined for sparse probability distributions. Furthermore, many sparse
normalization functions often collapse the multimodality of distributions. In
this work, we present $\textit{ev-softmax}$, a sparse normalization function
that preserves the multimodality of probability distributions. We derive its
properties, including its gradient in closed-form, and introduce a continuous
family of approximations to $\textit{ev-softmax}$ that have full support and
can be trained with probabilistic loss functions such as negative
log-likelihood and Kullback-Leibler divergence. We evaluate our method on a
variety of generative models, including variational autoencoders and
auto-regressive architectures. Our method outperforms existing dense and sparse
normalization techniques in distributional accuracy. We demonstrate that
$\textit{ev-softmax}$ successfully reduces the dimensionality of probability
distributions while maintaining multimodality.
