关于非对称核的行列式点过程
On determinantal point processes with nonsymmetric kernels
AI总结：
本文利用$P_0$矩阵理论给出非对称核行列式点过程良定义的必要充分条件，并推广常见结果，进而构造对称核正则DPP的吸引耦合以建模异标记点间的吸引。
摘要：
行列式点过程（简称DPP）是一类排斥点过程。它们在统计中用于建模具有近距离排斥性的空间点模式数据集。在有限集上的DPP中，它们由一个称为DPP核的矩阵定义，该矩阵通常假设为对称的。虽然存在一些非对称核的DPP例子，但关于这对它们通常性质的影响知之甚少。在本文中，我们展示了如何将关于$P_0$矩阵的结果适应到DPP设置中，以获得非对称核DPP良定义的充分必要条件。我们还推广了DPP上的各种常见结果。然后，我们展示了如何利用这些结果构造具有对称核的正则DPP的吸引耦合，以建模具有相同标记点之间排斥和不同标记点之间吸引的空间标记点模式。
英文摘要：
Determinantal point processes (DPPs for short) are a class of repulsive point processes. They have found some statistical applications to model spatial point pattern datasets with repulsion between close points. In the case of DPPs on finite sets, they are defined by a matrix called the DPP kernel which is usually assumed to be symmetric. While there are a few known examples of DPPs with nonsymmetric kernels, not much is known on how this affects their usual properties. In this paper, we demonstrate how to adapt the results on $P_0$ matrices to the DPP setting in order to get necessary and sufficient conditions for the well-definedness of DPPs with nonsymmetric kernels. We also generalize various common results on DPPs. We then show how to use these results to construct attractive couplings of regular DPPs with symmetric kernels in order to model spatial marked point patterns with repulsion between points of the same mark and attraction between points of different marks.
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