Linear causal disentanglement is a recent method in causal representation learning to describe a collection of observed variables via latent variables with causal dependencies between them. It can be viewed as a generalization of both independent component analysis and linear structural equation models. We study the identifiability of linear causal disentanglement, assuming access to data under multiple contexts, each given by an intervention on a latent variable. We show that one perfect intervention on each latent variable is sufficient and in the worst case necessary to recover parameters under perfect interventions, generalizing previous work to allow more latent than observed variables. We give a constructive proof that computes parameters via a coupled tensor decomposition. For soft interventions, we find the equivalence class of latent graphs and parameters that are consistent with observed data, via the study of a system of polynomial equations. Our results hold assuming the existence of non-zero higher-order cumulants, which implies non-Gaussianity of variables.

翻译：线性因果解耦是因果表示学习中的一种新方法，它通过具有因果依赖关系的潜变量来描述一组观测变量。它可以被视为独立成分分析和线性结构方程模型的推广。我们研究了线性因果解耦的可识别性，假设可以获得多种干预背景下的数据，每种背景由对某个潜变量的一次干预给定。我们证明，在完美干预条件下，对每个潜变量进行一次完美干预足以恢复参数，并且在最坏情况下也是必要的，从而将先前工作推广到允许潜变量数量多于观测变量的情形。我们给出了一个构造性证明，通过耦合张量分解来计算参数。对于软干预，我们通过研究一个多项式方程组，找到了与观测数据一致的潜变量图及参数的等价类。我们的结果建立在存在非零高阶累积量的假设之上，这蕴含了变量的非高斯性。