On the Convex Combination of Determinantal Point Processes

Determinantal point processes (DPPs) are attractive probabilistic models for expressing item quality and set diversity simultaneously. Although DPPs are widely-applicable to many subset selection tasks, there exist simple small-size probability distributions that any DPP cannot express. To overcome this drawback while keeping good properties of DPPs, in this paper we investigate the expressive power of \emph{convex combinations of DPPs}. We provide upper and lower bounds for the number of DPPs required for \emph{exactly} expressing any probability distribution. For the \emph{approximation} error, we give an upper bound on the Kullback–Leibler divergence $n-\lfloor \log t\rfloor +\epsilon$ for any $\epsilon >0$ of approximate distribution from a given joint probability distribution, where $t$ is the number of DPPs. Our numerical simulation on an online retail dataset empirically verifies that a convex combination of only two DPPs can outperform a nonsymmetric DPP in terms of the Kullback–Leibler divergence. By combining a polynomial number of DPPs, we can express probability distributions induced by bounded-degree pseudo-Boolean functions, which include weighted coverage functions of bounded occurrence.