Beyond identifiability: Learning causal representations with few environments and finite samples

We provide explicit, finite-sample guarantees for learning causal representations from data with a sublinear number of environments. Causal representation learning seeks to provide a rigourous foundation for the general representation learning problem by bridging causal models with latent factor models in order to learn interpretable representations with causal semantics. Despite a blossoming theory of identifiability in causal representation learning, estimation and finite-sample bounds are less well understood. We show that causal representations can be learned with only a logarithmic number of unknown, multi-node interventions, and that the intervention targets need not be carefully designed in advance. Through a careful perturbation analysis, we provide a new analysis of this problem that guarantees consistent recovery of (a) the latent causal graph, (b) the mixing matrix and representations, and (c) \emph{unknown} intervention targets.

💡 Research Summary

The paper tackles a central gap in causal representation learning (CRL): while many recent works have established identifiability of latent causal factors under multiple environments, they provide little guidance on how to actually estimate these factors from finite data and what sample complexity is required. The authors focus on a linear setting where high‑dimensional observations X∈ℝ^p (p≫d) are generated by a linear mixing X = B Z, with Z∈ℝ^d obeying a linear structural equation model Z = Aᵀ Z + ν. The matrix A encodes a directed acyclic graph (DAG) G over the latent variables, B is an unknown full‑column‑rank “decoder”, and ν is an independent noise vector with no restrictive distributional assumptions.

Crucially, the data are collected under K + 1 environments. In each environment k≥1 a (unknown) subset I(k)⊂