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Many applications of generative models rely on the marginalization of theirhigh-dimensional output probability distributions. Normalization functions thatyield sparse probability distributions can make exact marginalization morecomputationally tractable. However, sparse normalization functions usuallyrequire alternative loss functions for training since the log-likelihood isundefined for sparse probability distributions. Furthermore, many sparsenormalization functions often collapse the multimodality of distributions. Inthis work, we present $\textit{ev-softmax}$, a sparse normalization functionthat preserves the multimodality of probability distributions. We derive itsproperties, including its gradient in closed-form, and introduce a continuousfamily of approximations to $\textit{ev-softmax}$ that have full support andcan be trained with probabilistic loss functions such as negativelog-likelihood and Kullback-Leibler divergence. We evaluate our method on avariety of generative models, including variational autoencoders andauto-regressive architectures. Our method outperforms existing dense and sparsenormalization techniques in distributional accuracy. We demonstrate that$\textit{ev-softmax}$ successfully reduces the dimensionality of probabilitydistributions while maintaining multimodality.

Evidential Softmax for Sparse Multimodal Distributions in Deep Generative Models