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

TL;DR

This paper develops a rigorous mathematical framework for hybrid discrete-continuous representations in machine learning, enabling better interpretability and learning of discrete latent variables while maintaining differentiability.

Contribution

It introduces a new base measure, entropy function, and automaton theory to formalize hybrid discrete-continuous models in a mathematically rigorous way.

Findings

Established a new base measure on the face lattice of the probability simplex.

Defined a generalized entropy function encompassing discrete and differential entropies.

Developed a mixed weighted finite state automaton for recognizing hybrid languages.

Abstract

Neural networks and other machine learning models compute continuous representations, while humans communicate with discrete symbols. Reconciling these two forms of communication is desirable to generate human-readable interpretations or to learn discrete latent variable models, while maintaining end-to-end differentiability. Some existing approaches (such as the Gumbel-softmax transformation) build continuous relaxations that are discrete approximations in the zero-temperature limit, while others (such as sparsemax transformations and the hard concrete distribution) produce discrete/continuous hybrids. In this paper, we build rigorous theoretical foundations for these hybrids. Our starting point is a new "direct sum" base measure defined on the face lattice of the probability simplex. From this measure, we introduce a new entropy function that includes the discrete and differential entropies as particular cases, and has an interpretation in terms of code optimality, as well as two other information-theoretic counterparts that generalize the mutual information and Kullback-Leibler divergences. Finally, we introduce "mixed languages" as strings of hybrid symbols and a new mixed weighted finite state automaton that recognizes a class of regular mixed languages, generalizing closure properties of regular languages.