扩展L-系综：行列式点过程的一种新表示
Extended L-ensembles: a new representation for Determinantal Point Processes
摘要：
行列式点过程（DPPs）是一类排斥点过程，因其相对简单而流行。它们传统上是通过它们的边缘分布来定义的，但是被称为“L-系综”的DPP子集具有易处理的似然性，因此特别容易处理。事实上，在许多应用中，DPP更自然地基于L-系综公式而不是通过边缘核来定义。不幸的是，并非所有DPP都是L-系综，但有一个统一的描述。我们在这里引入扩展的L-系综，并表明所有的DPP是扩展的L-系综（反之亦然）。扩展的L-系综具有非常简单的似然函数，包含L-系综和投影DPP作为特例。从理论的角度来看，他们修复了DPP通常形式主义中的一些病态，例如投影DPP不是L-系综的事实。从实用的角度来看，他们扩展了一组核函数，可用于定义DPP：我们表明，条件正定核是很好的候选人定义DPP，包括DPP，不需要空间尺度参数。最后，扩展的L-系综是基于所谓的“鞍点矩阵”，我们证明了柯西-比奈定理的扩展，这样的矩阵，可能是独立的利益。
英文摘要：
Determinantal point processes (DPPs) are a class of repulsive point processes, popular for their relative simplicity. They are traditionally defined via their marginal distributions, but a subset of DPPs called "L-ensembles" have tractable likelihoods and are thus particularly easy to work with. Indeed, in many applications, DPPs are more naturally defined based on the L-ensemble formulation rather than through the marginal kernel. The fact that not all DPPs are L-ensembles is unfortunate, but there is a unifying description. We introduce here extended L-ensembles, and show that all DPPs are extended L-ensembles (and vice-versa). Extended L-ensembles have very simple likelihood functions, contain L-ensembles and projection DPPs as special cases. From a theoretical standpoint, they fix some pathologies in the usual formalism of DPPs, for instance the fact that projection DPPs are not L-ensembles. From a practical standpoint, they extend the set of kernel functions that may be used to define DPPs: we show that conditional positive definite kernels are good candidates for defining DPPs, including DPPs that need no spatial scale parameter. Finally, extended L-ensembles are based on so-called ``saddle-point matrices'', and we prove an extension of the Cauchy-Binet theorem for such matrices that may be of independent interest.
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