  Counterfactual generation lies at the core of various machine learning tasks,
including image translation and controllable text generation. This generation
process usually requires the identification of the disentangled latent
representations, such as content and style, that underlie the observed data.
However, it becomes more challenging when faced with a scarcity of paired data
and labeling information. Existing disentangled methods crucially rely on
oversimplified assumptions, such as assuming independent content and style
variables, to identify the latent variables, even though such assumptions may
not hold for complex data distributions. For instance, food reviews tend to
involve words like tasty, whereas movie reviews commonly contain words such as
thrilling for the same positive sentiment. This problem is exacerbated when
data are sampled from multiple domains since the dependence between content and
style may vary significantly over domains. In this work, we tackle the
domain-varying dependence between the content and the style variables inherent
in the counterfactual generation task. We provide identification guarantees for
such latent-variable models by leveraging the relative sparsity of the
influences from different latent variables. Our theoretical insights enable the
development of a doMain AdapTive counTerfactual gEneration model, called
(MATTE). Our theoretically grounded framework achieves state-of-the-art
performance in unsupervised style transfer tasks, where neither paired data nor
style labels are utilized, across four large-scale datasets. Code is available
at https://github.com/hanqi-qi/Matte.git
