Partitioning a set of elements into an unknown number of mutually exclusive subsets is essential in many machine learning problems.

Partitioning a set of elements into an unknown number of mutually exclusive subsets is essential in many machine learning problems.

Partitioning a set of elements into an unknown number of mutually exclusive subsets is essential in many machine learning problems. However, assigning elements, such as samples in a dataset or neurons in a network layer, to an unknown and discrete number of subsets is inherently non-differentiable, prohibiting end-to-end gradient-based optimization of parameters. We overcome this limitation by proposing a novel two-step method for inferring partitions, which allows its usage in variational inference tasks. This new approach enables reparameterized gradients with respect to the parameters of the new random partition model. Our method works by inferring the number of elements per subset and, second, by filling these subsets in a learned order. We highlight the versatility of our general-purpose approach on three different challenging experiments: variational clustering, inference of shared and independent generative factors under weak supervision, and multitask learning.

Random partition models are widely used in Bayesian methods for various clustering tasks, such as mixture models, topic models, and community detection problems. While the number of clusters induced by random partition models has been studied extensively, another important model property regarding the balancedness of cluster sizes has been largely neglected. We formulate a framework to define and theoretically study the balancedness of exchangeable random partition models, by analyzing how a model assigns probabilities to partitions with different levels of balancedness. We demonstrate that the "rich-get-richer" characteristic of many existing popular random partition models is an inevitable consequence of two common assumptions: product-form exchangeability and projectivity. We propose a principled way to compare the balancedness of random partition models, which gives a better understanding of what model works better and what doesn't for different applications. We also introduce the "rich-get-poorer" random partition models and illustrate their application to entity resolution tasks.

Many popular random partition models, such as the Chinese restaurant process and its two-parameter extension, fall in the class of exchangeable random partitions, and have found wide applicability in model-based clustering, population genetics, ecology or network analysis. While the exchangeability assumption is sensible in many cases, it has some strong implications. In particular, Kingman's representation theorem implies that the size of the clusters necessarily grows linearly with the sample size; this feature may be undesirable for some applications, as recently pointed out by Miller et al. (2015). We present here a flexible class of non-exchangeable random partition models which are able to generate partitions whose cluster sizes grow sublinearly with the sample size, and where the growth rate is controlled by one parameter. Along with this result, we provide the asymptotic behaviour of the number of clusters of a given size, and show that the model can exhibit a power-law behavior, controlled by another parameter. The construction is based on completely random measures and a Poisson embedding of the random partition, and inference is performed using a Sequential Monte Carlo algorithm. Additionally, we show how the model can also be directly used to generate sparse multigraphs with power-law degree distributions and degree sequences with sublinear growth. Finally, experiments on real datasets emphasize the usefulness of the approach compared to a two-parameter Chinese restaurant process.