  Determinantal point processes (DPPs) are popular probabilistic models that
arise in many machine learning tasks, where distributions of diverse sets are
characterized by matrix determinants. In this paper, we develop fast algorithms
to find the most likely configuration (MAP) of large-scale DPPs, which is
NP-hard in general. Due to the submodular nature of the MAP objective, greedy
algorithms have been used with empirical success. Greedy implementations
require computation of log-determinants, matrix inverses or solving linear
systems at each iteration. We present faster implementations of the greedy
algorithms by utilizing the complementary benefits of two log-determinant
approximation schemes: (a) first-order expansions to the matrix log-determinant
function and (b) high-order expansions to the scalar log function with
stochastic trace estimators. In our experiments, our algorithms are orders of
magnitude faster than their competitors, while sacrificing marginal accuracy.
