Revealing Hidden Causal Variables and Latent Factors from Multiple Distributions
Primary Area: causal reasoning
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Keywords: Identifiability, Latent Variable Models, Causal Representation Learning
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Abstract: 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 prediction in changing environments or making proper changes to the system, it is helpful to recover the hidden causal variables $Z_i$, their causal relations represented by graph $\mathcal{G}_Z$, and how their causal influences change, which can be explained by suitable latent factors $\theta_i$ governing changes in the causal mechanisms. This paper is concerned with the problem of estimating the underlying hidden causal variables and the latent factors from multiple distributions (arising from heterogeneous data or nonstationary time series) in nonparametric settings. We first show that under the sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, the recovered latent variables and their relations are related to the underlying causal model in a specific, nontrivial way. Moreover, we show that orthogonally, under the modular change condition on the causal modules (without the sparsity constraint on the graph), the underlying latent factors $\theta_i$ can be recovered up to component-wise invertible transformations. Putting them together, one is able to recover the hidden variables, their causal relations, and the corresponding latent factors up to minor indeterminacies.
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Submission Number: 444
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