  Determinantal Point Processes (DPPs) are a widely used probabilistic model
for negatively correlated sets. DPPs have been successfully employed in Machine
Learning applications to select a diverse, yet representative subset of data.
In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD
Thesis (2011) that the problem of finding a maximum likelihood DPP model for a
given data set is NP-complete.
  In this work we prove Kulesza's conjecture. In fact, we prove the following
stronger hardness of approximation result: even computing a
$\left(1-O(\frac{1}{\log^9{N}})\right)$-approximation to the maximum
log-likelihood of a DPP on a ground set of $N$ elements is NP-complete. At the
same time, we also obtain the first polynomial-time algorithm that achieves a
nontrivial worst-case approximation to the optimal log-likelihood: the
approximation factor is $\frac{1}{(1+o(1))\log{m}}$ unconditionally (for data
sets that consist of $m$ subsets), and can be improved to $1-\frac{1+o(1)}{\log
N}$ if all $N$ elements appear in a $O(1/N)$-fraction of the subsets.
  In terms of techniques, we reduce approximating the maximum log-likelihood of
DPPs on a data set to solving a gap instance of a "vector coloring" problem on
a hypergraph. Such a hypergraph is built on a bounded-degree graph construction
of Bogdanov, Obata and Trevisan (FOCS 2002), and is further enhanced by the
strong expanders of Alon and Capalbo (FOCS 2007) to serve our purposes.
