Nonparametric estimation of continuous DPPs with kernel methods
This work addresses a theoretical gap in statistical modeling for repulsive point patterns, offering a nonparametric inference method that could benefit machine learning and spatial statistics applications, though it appears incremental as it builds on existing representer theorems.
The authors tackled the open problem of nonparametric maximum likelihood estimation for continuous Determinantal Point Processes (DPPs) by showing that a restricted version of this optimization problem can be formulated as a finite-dimensional one using kernel methods, and they proposed a fixed-point algorithm to solve it while providing interpretable kernel estimates.
Determinantal Point Process (DPPs) are statistical models for repulsive point patterns. Both sampling and inference are tractable for DPPs, a rare feature among models with negative dependence that explains their popularity in machine learning and spatial statistics. Parametric and nonparametric inference methods have been proposed in the finite case, i.e. when the point patterns live in a finite ground set. In the continuous case, only parametric methods have been investigated, while nonparametric maximum likelihood for DPPs -- an optimization problem over trace-class operators -- has remained an open question. In this paper, we show that a restricted version of this maximum likelihood (MLE) problem falls within the scope of a recent representer theorem for nonnegative functions in an RKHS. This leads to a finite-dimensional problem, with strong statistical ties to the original MLE. Moreover, we propose, analyze, and demonstrate a fixed point algorithm to solve this finite-dimensional problem. Finally, we also provide a controlled estimate of the correlation kernel of the DPP, thus providing more interpretability.