  Despite the multifaceted recent advances in interventional causal
representation learning (CRL), they primarily focus on the stylized assumption
of single-node interventions. This assumption is not valid in a wide range of
applications, and generally, the subset of nodes intervened in an
interventional environment is fully unknown. This paper focuses on
interventional CRL under unknown multi-node (UMN) interventional environments
and establishes the first identifiability results for general latent causal
models (parametric or nonparametric) under stochastic interventions (soft or
hard) and linear transformation from the latent to observed space.
Specifically, it is established that given sufficiently diverse interventional
environments, (i) identifiability up to ancestors is possible using only soft
interventions, and (ii) perfect identifiability is possible using hard
interventions. Remarkably, these guarantees match the best-known results for
more restrictive single-node interventions. Furthermore, CRL algorithms are
also provided that achieve the identifiability guarantees. A central step in
designing these algorithms is establishing the relationships between UMN
interventional CRL and score functions associated with the statistical models
of different interventional environments. Establishing these relationships also
serves as constructive proof of the identifiability guarantees.
