This paper introduces a sparsity principle for causal representation learning in partially observable settings. The goal is to recover high-level causal variables from unstructured, high-dimensional observations, where each observation only captures a subset of the underlying causal state. The authors propose two theoretical results for identifying causal variables up to permutation and element-wise transformation under partial observability. The first result applies to linear mixing functions without parametric assumptions on the underlying causal model, while the second applies to piecewise linear mixing functions with Gaussian latent variables. Based on these insights, the authors propose two methods for estimating the underlying causal variables by enforcing sparsity in the inferred representation. Experiments on simulated datasets and established benchmarks demonstrate the effectiveness of their approach in recovering the ground-truth latent variables. The paper also discusses the implications of partial observability in real-world applications, such as when not all aspects of the environment can be observed at once. The authors highlight the importance of sparsity constraints in enabling identifiability in partially observable settings and show that their methods can be applied to both linear and piecewise linear mixing functions. The results demonstrate that their approach can recover causal variables even when only a subset of the underlying causal state is observed. The paper also discusses the limitations of their approach, including the need for group information in certain cases and the difficulty of satisfying Gaussianity constraints in practice. Overall, the work contributes to the field of causal representation learning by providing a new framework for learning causal variables in partially observable settings.This paper introduces a sparsity principle for causal representation learning in partially observable settings. The goal is to recover high-level causal variables from unstructured, high-dimensional observations, where each observation only captures a subset of the underlying causal state. The authors propose two theoretical results for identifying causal variables up to permutation and element-wise transformation under partial observability. The first result applies to linear mixing functions without parametric assumptions on the underlying causal model, while the second applies to piecewise linear mixing functions with Gaussian latent variables. Based on these insights, the authors propose two methods for estimating the underlying causal variables by enforcing sparsity in the inferred representation. Experiments on simulated datasets and established benchmarks demonstrate the effectiveness of their approach in recovering the ground-truth latent variables. The paper also discusses the implications of partial observability in real-world applications, such as when not all aspects of the environment can be observed at once. The authors highlight the importance of sparsity constraints in enabling identifiability in partially observable settings and show that their methods can be applied to both linear and piecewise linear mixing functions. The results demonstrate that their approach can recover causal variables even when only a subset of the underlying causal state is observed. The paper also discusses the limitations of their approach, including the need for group information in certain cases and the difficulty of satisfying Gaussianity constraints in practice. Overall, the work contributes to the field of causal representation learning by providing a new framework for learning causal variables in partially observable settings.