  In this technical report, we discuss several sampling algorithms for
Determinantal Point Processes (DPP). DPPs have recently gained a broad interest
in the machine learning and statistics literature as random point processes
with negative correlation, i.e., ones that can generate a "diverse" sample from
a set of items. They are parametrized by a matrix $\mathbf{L}$, called
$L$-ensemble, that encodes the correlations between items. The standard
sampling algorithm is separated in three phases: 1/~eigendecomposition of
$\mathbf{L}$, 2/~an eigenvector sampling phase where $\mathbf{L}$'s
eigenvectors are sampled independently via a Bernoulli variable parametrized by
their associated eigenvalue, 3/~a Gram-Schmidt-type orthogonalisation procedure
of the sampled eigenvectors.
  In a naive implementation, the computational cost of the third step is on
average $\mathcal{O}(N\mu^3)$ where $\mu$ is the average number of samples of
the DPP. We give an algorithm which runs in $\mathcal{O}(N\mu^2)$ and is
extremely simple to implement. If memory is a constraint, we also describe a
dual variant with reduced memory costs. In addition, we discuss implementation
details often missing in the literature.
