Scaled subordinators and generalizations of the Indian buffet process

Abstract

We study random families of subsets of $\mathbb{N}$ that are similar to exchangeable random partitions, but do not require constituent sets to be disjoint: Each element of $\mathbb{N}$ may be contained in multiple subsets. One class of such objects, known as Indian buffet processes, has become a popular tool in machine learning. Based on an equivalence between Indian buffet and scale-invariant Poisson processes, we identify a random scaling variable whose role is similar to that played in exchangeable partition models by the total mass of a random measure. Analogous to the construction of exchangeable partitions from normalized subordinators, random families of sets can be constructed from randomly scaled subordinators. Coupling to a heavy-tailed scaling variable induces a power law on the number of sets containing the first $n$ elements. Several examples, with properties desirable in applications, are derived explicitly. A relationship to exchangeable partitions is made precise as a correspondence between scaled subordinators and Poisson-Kingman measures, generalizing a result of Arratia, Barbour and Tavare on scale-invariant processes.

References in corpus (2)

• Clustered Multi-Task Learning: A Convex Formulation
• The combinatorial structure of beta negative binomial processes

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