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Determinantal point processes (DPPs) are an elegant model for encodingprobabilities over subsets, such as shopping baskets, of a ground set, such asan item catalog. They are useful for a number of machine learning tasks,including product recommendation. DPPs are parametrized by a positivesemi-definite kernel matrix. Recent work has shown that using a low-rankfactorization of this kernel provides remarkable scalability improvements thatopen the door to training on large-scale datasets and computing onlinerecommendations, both of which are infeasible with standard DPP models that usea full-rank kernel. In this paper we present a low-rank DPP mixture model thatallows us to represent the latent structure present in observed subsets as amixture of a number of component low-rank DPPs, where each component DPP isresponsible for representing a portion of the observed data. The mixture modelallows us to effectively address the capacity constraints of the low-rank DPPmodel. We present an efficient and scalable Markov Chain Monte Carlo (MCMC)learning algorithm for our model that uses Gibbs sampling and stochasticgradient Hamiltonian Monte Carlo (SGHMC). Using an evaluation on severalreal-world product recommendation datasets, we show that our low-rank DPPmixture model provides substantially better predictive performance than ispossible with a single low-rank or full-rank DPP, and significantly betterperformance than several other competing recommendation methods in many cases.

The Bayesian Low-Rank Determinantal Point Process Mixture Model