Nystrom Approximation for Large-Scale Determinantal Processes

Determinantal point processes (DPPs) are appealing models for subset selection problems where diversity is desired. They offer surprisingly efficient inference, including sampling in O(N) time and O(N) space, where N is the number of base items. However, in some applications, N may grow so large that sampling from a DPP becomes computationally infeasible. This is especially true in settings where the DPP kernel matrix cannot be represented by a linear decomposition of low-dimensional feature vectors. In these cases, we propose applying the Nyström approximation to project the kernel matrix into a low-dimensional space. While theoretical guarantees for the Nyström approximation in terms of standard matrix norms have been previously established, we are concerned with probabilistic measures, like total variation distance between the DPP and its Nyström approximation, that behave quite differently. In this paper we derive new error bounds for the Nyström-approximated DPP and present empirical results to corroborate them. We then demonstrate the Nyström-approximated DPP by applying it to a motion capture summarization task.