学习绘制在概率度量空间中的随机几何图

中文摘要 AI 辅助

我们提出了一种针对多变量数据集的数据驱动型随机几何图（Random Geometric Graph, RGG）学习新方法，该图绘制在概率度量空间中。这种图学习方法适用于各类通用数据集，与观测值的类型、概率分布或数据规模无关。我们将图所绘制空间的度量定义为所引入的随机变量的概率分布，该变量表示图中两个顶点的连通性差异，以及分别附着于这两个顶点的两个随机变量之间的相关性。我们提出该差异变量的累积分布函数（cumulative distribution function, cdf）作为所学习RGG的宿主空间的距离函数，当任意两个节点间的距离小于所选的截断概率时，这两个节点之间存在边。将RGG绘制在该概率空间中会使该图成为软随机几何图（Soft RGG），即若边存在，则其以确定的概率存在。我们提出一种简单的基于拒绝采样（Rejection Sampling）的技术来学习任意边的概率。该RGG顶点的期望度分布被确定为局部的，且依赖于观测值间的相关矩阵。若该相关矩阵未知，则可利用我们提出的闭式后验概率密度函数从数据中学习得到。我们通过对高度多变量的真实数据集学习多个RGG，来说明所提出的图学习方法。

英文摘要

We present a new data-driven learning of a Random Geometric Graph (RGG) of a multivariate dataset, where the graph is drawn in a probabilistic metric space. This graph learning works for generic datasets, irrespective of the type of the observables; their probability distributions; or size of the data. We identify a metric of the space that the graph is drawn in, as a probability distribution of a random variable that we introduce, namely, a variable that represents the disparity between the connectedness of two vertices of the graph, and the correlation between the two random variables that are attached to the respective vertex. It is the closed-form {\it{cdf}} of this disparity variable that we advance as the distance function of the host space of the learnt RGG, such that the edge exists between any two nodes, if this inter-nodal distance falls short of a chosen cutoff probability. Drawing the RGG in this probabilistic space leads to the graph being an Soft RGG, such that any edge - if it exists - exists with an identified probability. We forward a simple Rejection Sampling-based technique for learning the probability of any edge. The expected degree distribution of a vertex of this RGG is identified as local, and dependent on the inter-observable correlation matrix. If said correlation matrix is not known, it can be learnt given the data, using its closed-form posterior probability density function, that we forward. We illustrate our graph learning method by learning multiple RGGs of highly multivariate real datasets.

发表机构

• University of York（约克大学）
• Alan Turing Institute（阿兰·图灵研究所）

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