This paper studies the causal representation learning problem when the latent causal variables are observed indirectly through an unknown linear transformation. The objectives are: (i) recovering the unknown linear transformation (up to scaling) and (ii) determining the directed acyclic graph (DAG) underlying the latent variables. Sufficient conditions for DAG recovery are established, and it is shown that a large class of non-linear models in the latent space (e.g., causal mechanisms parameterized by two-layer neural networks) satisfy these conditions. These sufficient conditions ensure that the effect of an intervention can be detected correctly from changes in the score. Capitalizing on this property, recovering a valid transformation is facilitated by the following key property: any valid transformation renders latent variables' score function to necessarily have the minimal variations across different interventional environments. This property is leveraged for perfect recovery of the latent DAG structure using only \emph{soft} interventions. For the special case of stochastic \emph{hard} interventions, with an additional hypothesis testing step, one can also uniquely recover the linear transformation up to scaling and a valid causal ordering.

翻译：本文研究了潜在因果变量通过未知线性变换间接观测时的因果表示学习问题。目标包括：(i) 恢复未知线性变换（至尺度等价性）以及 (ii) 确定潜在变量背后的有向无环图（DAG）。本文建立了DAG恢复的充分条件，并表明潜在空间中的一大类非线性模型（例如，由两层神经网络参数化的因果机制）满足这些条件。这些充分条件确保了干预效应能够通过分值的变化被正确检测。基于这一特性，以下关键性质促进了有效变换的恢复：任何有效变换必然使潜在变量的分值函数在不同干预环境下具有最小变异。该性质被用于仅使用软干预实现潜在DAG结构的完美恢复。对于随机硬干预这一特例，结合额外的假设检验步骤，还可以唯一地恢复线性变换至尺度等价性以及有效的因果排序。