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.

翻译：磁点过程(DPPs)是令人厌恶的点模式的统计模型。对于DPP来说,取样和推断都是可移动的。DPP是具有负依赖性的模型中一个罕见的特征,说明其在机器学习和空间统计中很受欢迎。在有限的情况下,即当点模式生活在一个有限的地面组时,提出了参数和非参数推论方法。在连续的情况下,只调查了参数方法,而对于DPP来说,非参数的最大可能性 -- -- 微量级操作员的优化问题 -- -- 仍然是个未决问题。在本文中,我们表明,这一最大可能性(MLE)问题的一个限制性版本属于最近代表RKHS中非负性功能的理论的范围。这导致了一个与原始MLE有很强的统计联系的有限维问题。此外,我们提议、分析并展示一个固定点算法来解决这一有限维问题。最后,我们还提供了对DPP的关联核心的受控估计,从而提供了更多的解释性。