We consider graphical models based on a recursive system of linear structural equations. This implies that there is an ordering, $\sigma$, of the variables such that each observed variable $Y_v$ is a linear function of a variable specific error term and the other observed variables $Y_u$ with $\sigma(u) < \sigma (v)$. The causal relationships, i.e., which other variables the linear functions depend on, can be described using a directed graph. It has been previously shown that when the variable specific error terms are non-Gaussian, the exact causal graph, as opposed to a Markov equivalence class, can be consistently estimated from observational data. We propose an algorithm that yields consistent estimates of the graph also in high-dimensional settings in which the number of variables may grow at a faster rate than the number of observations, but in which the underlying causal structure features suitable sparsity; specifically, the maximum in-degree of the graph is controlled. Our theoretical analysis is couched in the setting of log-concave error distributions.

中文翻译：

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非高斯下的高维因果发现

我们考虑基于线性结构方程递归系统的图形模型。这意味着变量的排序 $\sigma$ 使得每个观察变量 $Y_v$ 是变量特定误差项的线性函数，其他观察变量 $Y_u$ 与 $\sigma(u) < \sigma (v)$。因果关系，即线性函数所依赖的其他变量，可以使用有向图来描述。之前已经表明，当变量特定误差项是非高斯时，精确的因果图，而不是马尔可夫等价类，可以从观测数据中一致地估计出来。我们提出了一种算法，该算法在高维设置中也能产生一致的图估计，其中变量的数量可能以比观察数量更快的速度增长，但其中潜在的因果结构具有适当的稀疏性；具体来说，控制图形的最大入度。我们的理论分析基于对数凹面误差分布的设置。