k-DPPs: Fixed-Size Determinantal Point Processes

Determinantal point processes (DPPs) have recently been proposed as models for set selection problems where diversity is preferred. For example, they can be used to select diverse sets of sentences to form document summaries, or to find multiple nonoverlapping human poses in an image. However, DPPs conflate the modeling of two distinct characteristics: the size of the set, and its content. For many realistic tasks, the size of the desired set is known up front; e.g., in search we may want to show the user exactly ten results. In these situations the effort spent by DPPs modeling set size is not only wasteful, but actually introduces unwanted bias into the modeling of content. Instead, we propose the k-DPP, a conditional DPP that models only sets of cardinality k. In exchange for this restriction, k-DPPs offer greater expressiveness and control over content, and simplified integration into applications like search. We derive algorithms for efficiently normalizing, sampling, and marginalizing kDPPs, and propose an experts-style algorithm for learning combinations of k-DPPs. We demonstrate the usefulness of the model on an image search task, where k-DPPs significantly outperform MMR as judged by human annotators.