通过邻域晶格学习通用的条件独立结构
Learning general conditional independence structures via the neighbourhood lattice
AI总结：
本文提出通过邻域晶格分解学习高维条件独立结构，无需先验图知识，有效避免维度诅咒。
摘要：
我们研究了非参数和高维环境下学习多变量依赖结构的问题，包括但不限于图模型。我们的方法结合了之前工作缺失的特征：展示如何非参数化学习整个依赖结构，同时避免维度诅咒并放松如忠实性等常见假设。为此，我们引入并研究了分布的邻域晶格分解，这是一种紧凑且非图表示的条件独立性（CI）表示，在无忠实图表示时有效。我们证明邻域晶格分解存在于任何图模型中，并且可以在高维空间中高效、非参数化和一致地计算。这为学习任何图模型所隐含的所有独立关系提供了方法，无需先验图知识或图类型。作为特殊情况，我们的结果为任何图模型上高维非参数估计条件独立结构提供了通用解决方案。
英文摘要：
We study the problem of learning multivariate dependencies in nonparametric and high-dimensional settings. This includes but is not limited to graphical models. Our approach effectively combines several features that are missing from previous work on this problem: We show how the entire dependence structure can be learned nonparametrically while simultaneously evading the curse of dimensionality and relaxing common assumptions such as faithfulness. To this end, we introduce and study the neighbourhood lattice decomposition of a distribution, which is a compact, non-graphical representation of conditional independence (CI) that is valid in the absence of a faithful graphical representation. We show that the neighbourhood lattice decomposition exists in any graphical model and can be computed efficiently, nonparametrically, and consistently in high-dimensions without paying the usual curse of dimensionality. This gives a way to learn all of the independence relations implied by any graphical model, without requiring a priori knowledge of the graph or even the graph type. As a special case, our results provide a general solution to the problem of nonparametric estimation of high-dimensional CI structures over any graphical model.
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