  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
k- DPPs on well-conditioned kernels. Our experiments show that this improvement
is significant for larger values of k, supporting our theoretical result.
