The continuous categorical: a novel simplex-valued exponential family

TL;DR

This paper introduces the continuous categorical, a new exponential family distribution for simplex-valued data, addressing limitations of Dirichlet models by providing unbiased estimators and better numerical stability, with demonstrated empirical improvements.

Contribution

The paper proposes the continuous categorical distribution, a novel exponential family for modeling simplex data, overcoming biases and numerical issues of traditional models like the Dirichlet.

Findings

Outperforms Dirichlet in simulations and real-world tasks

Provides unbiased estimators and reparameterization-friendly sampling methods

Demonstrates improved performance in neural network compression and election data analysis

Abstract

Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other related loss functions; here we show these standard choices suffer systematically from a number of limitations, including bias and numerical issues that frustrate the use of flexible network models upstream of these distributions. We resolve these limitations by introducing a novel exponential family of distributions for modeling simplex-valued data - the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli. Unlike the Dirichlet and other typical choices, the continuous categorical results in a well-behaved probabilistic loss function that produces unbiased estimators, while preserving the mathematical simplicity of the Dirichlet. As well as exploring its theoretical properties, we introduce sampling methods for this distribution that are amenable to the reparameterization trick, and evaluate their performance. Lastly, we demonstrate that the continuous categorical outperforms standard choices empirically, across a simulation study, an applied example on multi-party elections, and a neural network compression task.