  We study the task of determinant maximization under partition constraint, in
the context of large data sets. Given a point set $V\subset \mathbb{R}^d$ that
is partitioned into $s$ groups $V_1,..., V_s$, and integers $k_1,...,k_s$ where
$k=\sum_i k_i$, the goal is to pick $k_i$ points from group $i$ such that the
overall determinant of the picked $k$ points is maximized. Determinant
Maximization and its constrained variants have gained a lot of interest for
modeling diversityand have found applications in the context of fairness and
data summarization.
  We study the design of composable coresets for the constrained determinant
maximization problem. Composable coresets are small subsets of the data that
(approximately) preserve optimal solutions to optimization tasks and enable
efficient solutions in several other large data models including the
distributed and the streaming settings. In this work, we consider two regimes.
For the case of $k>d$, we show a peeling algorithm that gives us a composable
coreset of size $kd$ with an approximation factor of $d^{O(d)}$. We complement
our results by showing that this approximation factor is tight. For the case of
$k\leq d$, we show that a simple modification of the previous algorithms
results in an optimal coreset verified by our lower bounds. Our results apply
to all strongly Rayleigh distribution and several other experimental design
problems. In addition, we show coreset construction algorithms under the more
general laminar matroid constraints.
