On Causal Representation Learning with Internal Auxiliaries

Causal representation learning typically achieves identifiability using auxiliary variables external to the mixing process. This fails when observable sources act as internal mixing inputs whose entanglement invalidates standard volume-change proofs. We introduce a framework that treats observable sources as internal auxiliaries. We establish identifiability when all observed internal sources are used as conditioning variables. Under a volume-preserving mixing assumption, suitable variability conditions, and additional alignment conditions on the learned model, the unobserved source subspaces are identifiable up to an Independent Subspace Analysis ({ISA})-style equivalence class—a permutation of conditionally independent subspaces with within-subspace invertible transformations. In practice, we approximate the volume-preserving restriction with encoders that stabilize the induced distortion. Leveraging the known causal graph, we further propose a scheme that selects conditioning sources so as to preserve fine-grained factorizations of the unobserved sources. Experiments show improved recovery over representative auxiliary-based baselines in settings where treating internal sources as external conditioners can induce cross-factor leakage.