  Cross-Domain Recommendation (CDR) seeks to enable effective knowledge
transfer across domains. Existing works rely on either representation alignment
or transformation bridges, but they struggle on identifying domain-shared from
domain-specific latent factors. Specifically, while CDR describes user
representations as a joint distribution over two domains, these methods fail to
account for its joint identifiability as they primarily fixate on the marginal
distribution within a particular domain. Such a failure may overlook the
conditionality between two domains and how it contributes to latent factor
disentanglement, leading to negative transfer when domains are weakly
correlated. In this study, we explore what should and should not be transferred
in cross-domain user representations from a causality perspective. We propose a
Hierarchical subspace disentanglement approach to explore the Joint
IDentifiability of cross-domain joint distribution, termed HJID, to preserve
domain-specific behaviors from domain-shared factors. HJID organizes user
representations into layers: generic shallow subspaces and domain-oriented deep
subspaces. We first encode the generic pattern in the shallow subspace by
minimizing the Maximum Mean Discrepancy of initial layer activation. Then, to
dissect how domain-oriented latent factors are encoded in deeper layers
activation, we construct a cross-domain causality-based data generation graph,
which identifies cross-domain consistent and domain-specific components,
adhering to the Minimal Change principle. This allows HJID to maintain
stability whilst discovering unique factors for different domains, all within a
generative framework of invertible transformations that guarantee the joint
identifiability. With experiments on real-world datasets, we show that HJID
outperforms SOTA methods on a range of strongly and weakly correlated CDR
tasks.
