Bayesian Low-Rank Determinantal Point Processes

Abstract

Determinantal point processes (DPPs) have attracted significant attention as an elegant model that is able to capture the balance between quality and diversity within sets. DPPs are parameterized by a positive semi-definite kernel matrix. While DPPs have substantial expressive power, they are fundamentally limited by the parameterization of the kernel matrix and their inability to capture nonlinear interactions between items within sets. We present the deep DPP model as way to address these limitations, by using a deep feed-forward neural network to learn the kernel matrix. In addition to allowing us to capture nonlinear item interactions, the deep DPP also allows easy incorporation of item metadata into DPP learning. Since the learning target is the DPP kernel matrix, the deep DPP allows us to use existing DPP algorithms for efficient learning, sampling, and prediction. Through an evaluation on several real-world datasets, we show experimentally that the deep DPP can provide a considerable improvement in the predictive performance of DPPs, while also outperforming strong baseline models in many cases.Preprint. Under review.

Personalized Bundle List Recommendation

“…Here, we use the beam search to prune the low-quality ones down to y cand with width M, which is crucial for the efficiency. The submodular assumption [16] of DPPs is widely used during inference in practice including recommendation systems [6,10], where one can find the solution in polynomial time. Following Equation (5), we choose the bundle b from y cand one by one maximizing P(y ∪ {b}), which is given by:…”

Tensorized Determinantal Point Processes for Recommendation

“…For instance, if similarity is defined using image descriptors, then images of differing appearance will be selected by a DPP. On the other hand, if the entries L ij are learned using previously observed sets, such as e-commerce baskets (Gartrell, Paquet, and Koenigstein 2017), then co-purchased items i and j are likely to be sampled by the DPP, and thus the "similarity" L ij will be low. In an application such as a search engine or in document summarization, the kernel may be defined using feature descriptors ψ i ∈ R D (i.e tf-idf of the text), and a relevance score q i ∈ R + of each item i such that L ij = q i ψ T i ψ j q j , which favors relevant items (large q i ) and discourages lists composed of similar items.…”

“…We assume that there exists a latent space such that diverse items in this space are likely to be purchased together. Similarly to (Gartrell, Paquet, and Koenigstein 2017), we introduce a low-rank factorization of the kernel matrix L ∈ R p×p :…”

“…Refs. (Gartrell, Paquet, and Koenigstein 2017;Gartrell, Paquet, and Koenigstein 2016) address this combinatorial problem using a model based on Determinantal Point Processes (DPPs) for basket completion. DPPs are elegant probabilistic models of repulsion from quantum physics, which are used for a variety of tasks in machine learning (Kulesza and Taskar 2012).…”

“…In this setting, the DPP is parameterized by a kernel matrix of size p × p, where p is the size of the catalog. Thus the number of parameters to fit grows quadratically with p, and the computational complexity for learning, prediction, and sampling grows cubicly with p. As learning a full-rank DPP is hard, (Gartrell, Paquet, and Koenigstein 2017) proposes regularizing the DPP by constraining the kernel to be low rank. This regularization also improves generalization and offers more diversity in recommendations, without hurting predictive performance.…”