  Determinantal point processes (DPPs) are widely popular probabilistic models
used in machine learning to capture diversity in random subsets of items. While
traditional DPPs are defined by a symmetric kernel matrix, recent work has
shown a significant increase in the modeling power and applicability of models
defined by nonsymmetric kernels, where the model can capture interactions that
go beyond diversity. We study the problem of maximum a posteriori (MAP)
inference for determinantal point processes defined by a nonsymmetric positive
semidefinite matrix (NDPPs), where the goal is to find the maximum $k\times k$
principal minor of the kernel matrix $L$. We obtain the first multiplicative
approximation guarantee for this problem using local search, a method that has
been previously applied to symmetric DPPs. Our approximation factor of
$k^{O(k)}$ is nearly tight, and we show theoretically and experimentally that
it compares favorably to the state-of-the-art methods for this problem that are
based on greedy maximization. The main new insight enabling our improved
approximation factor is that we allow local search to update up to two elements
of the solution in each iteration, and we show this is necessary to have any
multiplicative approximation guarantee.
