  A determinantal point process (DPP) is a probabilistic model of set diversity
compactly parameterized by a positive semi-definite kernel matrix. To fit a DPP
to a given task, we would like to learn the entries of its kernel matrix by
maximizing the log-likelihood of the available data. However, log-likelihood is
non-convex in the entries of the kernel matrix, and this learning problem is
conjectured to be NP-hard. Thus, previous work has instead focused on more
restricted convex learning settings: learning only a single weight for each row
of the kernel matrix, or learning weights for a linear combination of DPPs with
fixed kernel matrices. In this work we propose a novel algorithm for learning
the full kernel matrix. By changing the kernel parameterization from matrix
entries to eigenvalues and eigenvectors, and then lower-bounding the likelihood
in the manner of expectation-maximization algorithms, we obtain an effective
optimization procedure. We test our method on a real-world product
recommendation task, and achieve relative gains of up to 16.5% in test
log-likelihood compared to the naive approach of maximizing likelihood by
projected gradient ascent on the entries of the kernel matrix.
