In many problems, the measured variables (e.g., image pixels) are just mathematical functions of the hidden causal variables (e.g., the underlying concepts or objects). For the purpose of making predictions in changing environments or making proper changes to the system, it is helpful to recover the hidden causal variables $Z_i$ and their causal relations represented by graph $\mathcal{G}_Z$. This problem has recently been known as causal representation learning. This paper is concerned with a general, completely nonparametric setting of causal representation learning from multiple distributions (arising from heterogeneous data or nonstationary time series), without assuming hard interventions behind distribution changes. We aim to develop general solutions in this fundamental case; as a by product, this helps see the unique benefit offered by other assumptions such as parametric causal models or hard interventions. We show that under the sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, interestingly, one can recover the moralized graph of the underlying directed acyclic graph, and the recovered latent variables and their relations are related to the underlying causal model in a specific, nontrivial way. In some cases, each latent variable can even be recovered up to component-wise transformations. Experimental results verify our theoretical claims.

翻译：在许多问题中，观测变量（如图像像素）仅是隐式因果变量（如基础概念或物体）的数学函数。为在变化环境中进行预测或对系统进行适当修改，恢复隐式因果变量$Z_i$及其由图$\mathcal{G}_Z$表示的因果关系十分有益。这一问题近年来被称为因果表征学习。本文研究多分布（源于异构数据或非平稳时间序列）下完全非参数设定的因果表征学习一般情形，且不假设分布变化背后存在硬干预。我们旨在为这一基础情形开发通用解决方案；作为副产品，这有助于阐明参数化因果模型或硬干预等其他假设所能提供的独特优势。研究表明，在稀疏性约束下，通过恢复潜变量图并满足因果影响的充分变化条件，有趣的是，可以恢复底层有向无环图的道德图，且恢复的潜变量及其关系以特定非平凡的方式与底层因果模型相关联。在某些情形下，每个潜变量甚至可恢复至分量级变换。实验结果验证了我们的理论主张。