Causal discovery aims to recover causal structures or models underlying the observed data. Despite its success in certain domains, most existing methods focus on causal relations between observed variables, while in many scenarios the observed ones may not be the underlying causal variables (e.g., image pixels), but are generated by latent causal variables or confounders that are causally related. To this end, in this paper, we consider Linear, Non-Gaussian Latent variable Models (LiNGLaMs), in which latent confounders are also causally related, and propose a Generalized Independent Noise (GIN) condition to estimate such latent variable graphs. Specifically, for two observed random vectors $\mathbf{Y}$ and $\mathbf{Z}$, GIN holds if and only if $\omega^{\intercal}\mathbf{Y}$ and $\mathbf{Z}$ are statistically independent, where $\omega$ is a parameter vector characterized from the cross-covariance between $\mathbf{Y}$ and $\mathbf{Z}$. From the graphical view, roughly speaking, GIN implies that causally earlier latent common causes of variables in $\mathbf{Y}$ d-separate $\mathbf{Y}$ from $\mathbf{Z}$. Interestingly, we find that the independent noise condition, i.e., if there is no confounder, causes are independent from the error of regressing the effect on the causes, can be seen as a special case of GIN. Moreover, we show that GIN helps locate latent variables and identify their causal structure, including causal directions. We further develop a recursive learning algorithm to achieve these goals. Experimental results on synthetic and real-world data demonstrate the effectiveness of our method.

Weaknesses: - Some of the points I'll make here are more conceptual and would just like to hear from the authors what their thoughts are. However, there is another school of thought related to the Nonparanormal distribution that says everything can be transformed into something that looks Gaussian. In practice, either in applications or real-world analyses the authors have undertaken, what has been their experience in the usage of Gaussian vs non-Gaussian methods for structure learning. Is the first method they recommend a non-Gaussian method or one that relies on Gaussianity assumptions? In that sense, it is closer to algorithms like PC/FCI that are in theory nonparametric.

The paper introduces a structure learning method via a relationship between an algebraic condition and structures in a graphical model. The reviewers felt this was a novel and interesting contribution to the structure learning literature, and the experimental evaluation provided by the authors was found to be quite thorough.

Causal discovery aims to recover causal structures or models underlying the observed data. Despite its success in certain domains, most existing methods focus on causal relations between observed variables, while in many scenarios the observed ones may not be the underlying causal variables (e.g., image pixels), but are generated by latent causal variables or confounders that are causally related. To this end, in this paper, we consider Linear, Non-Gaussian Latent variable Models (LiNGLaMs), in which latent confounders are also causally related, and propose a Generalized Independent Noise (GIN) condition to estimate such latent variable graphs. Specifically, for two observed random vectors \mathbf{Y} and \mathbf{Z}, GIN holds if and only if \omega {\intercal}\mathbf{Y} and \mathbf{Z} are statistically independent, where \omega is a parameter vector characterized from the cross-covariance between \mathbf{Y} and \mathbf{Z} . From the graphical view, roughly speaking, GIN implies that causally earlier latent common causes of variables in \mathbf{Y} d-separate \mathbf{Y} from \mathbf{Z} .