Score-based Causal Representation Learning from Interventions: Nonparametric Identifiability

This paper focuses on causal representation learning (CRL) under a general nonparametric causal latent model and a general transformation model mapping the latent data to the observational data. It establishes identifiability and achievability results under two hard interventions per node in the latent causal graph, and one does not know which pair of environments have the same node intervened (uncoupled environments). Specifically, for identifiability, it is shown that perfect recovery of the latent causal model and variables is guaranteed under these conditions. For achievability, an algorithm is designed that uses observational data and two interventional environments per node and recovers the latent causal model and variables. This algorithm leverages score variations across different environments to estimate the inverse of the transformer and, subsequently, the latent variables. Our analysis also recovers the existing identifiability result for two hard interventions when metadata about the pair of environments that have the same node intervened is known (coupled environments). The existing results on non-parametric identifiability require assumptions on interventions and additional faithfulness assumptions. This paper shows that when observational data is available, additional assumptions on faithfulness are not necessary.