On Sampling and Greedy MAP Inference of Constrained Determinantal Point Processes

Subset selection problems ask for a small, diverse yet representative subset of the given data. When pairwise similarities are captured by a kernel, the determinants of submatrices provide a measure of diversity or independence of items within a subset. Matroid theory gives another notion of independence, thus giving rise to optimization and sampling questions about Determinantal Point Processes (DPPs) under matroid constraints. Partition constraints, as a special case, arise naturally when incorporating additional labeling or clustering information, besides the kernel, in DPPs. Finding the maximum determinant submatrix under matroid constraints on its row/column indices has been previously studied. However, the corresponding question of sampling from DPPs under matroid constraints has been unresolved, beyond the simple cardinality constrained k-DPPs. We give the first polynomial time algorithm to sample exactly from DPPs under partition constraints, for any constant number of partitions. We complement this by a complexity theoretic barrier that rules out such a result under general matroid constraints. Our experiments indicate that partition-constrained DPPs offer more flexibility and more diversity than k-DPPs and their naive extensions, while being reasonably efficient in running time. We also show that a simple greedy initialization followed by local search gives improved approximation guarantees for the problem of MAP inference from kDPPs on well-conditioned kernels. Our experiments show that this improvement is significant for larger values of k, supporting our theoretical result.