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.

翻译：我们研究在非参数和高维设定下学习多元依赖关系的问题。这包括但不限于图模型。我们的方法有效地结合了先前研究此问题的工作所缺失的若干特点：我们展示了如何非参数地学习整个依赖结构，同时规避维数灾难并放宽常见假设（如忠实性）。为此，我们引入并研究了分布的邻域格分解——这是一种在缺乏忠实图表示时仍有效的紧凑型非图式条件独立（CI）表示。我们证明，邻域格分解存在于任何图模型中，并且可以在高维情况下高效、非参数且一致地计算，而无需付出通常的维数代价。这提供了一种学习任何图模型所蕴含的所有独立关系的方法，且无需事先知道图结构甚至图类型。作为特例，我们的结果为任意图模型上高维CI结构的非参数估计问题提供了通用解决方案。