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Neural networks and other machine learning models compute continuousrepresentations, while humans communicate with discrete symbols. Reconcilingthese two forms of communication is desirable to generate human-readableinterpretations or to learn discrete latent variable models, while maintainingend-to-end differentiability. Some existing approaches (such as theGumbel-softmax transformation) build continuous relaxations that are discreteapproximations in the zero-temperature limit, while others (such as sparsemaxtransformations and the hard concrete distribution) produce discrete/continuoushybrids. In this paper, we build rigorous theoretical foundations for thesehybrids. Our starting point is a new "direct sum" base measure defined on theface lattice of the probability simplex. From this measure, we introduce a newentropy function that includes the discrete and differential entropies asparticular cases, and has an interpretation in terms of code optimality, aswell as two other information-theoretic counterparts that generalize the mutualinformation and Kullback-Leibler divergences. Finally, we introduce "mixedlanguages" as strings of hybrid symbols and a new mixed weighted finite stateautomaton that recognizes a class of regular mixed languages, generalizingclosure properties of regular languages.

Reconciling the Discrete-Continuous Divide: Towards a Mathematical Theory of Sparse Communication