  Generating data with properties of interest by external users while following
the right causation among its intrinsic factors is important yet has not been
well addressed jointly. This is due to the long-lasting challenge of jointly
identifying key latent variables, their causal relations, and their correlation
with properties of interest, as well as how to leverage their discoveries
toward causally controlled data generation. To address these challenges, we
propose a novel deep generative framework called the Correlation-aware Causal
Variational Auto-encoder (C2VAE). This framework simultaneously recovers the
correlation and causal relationships between properties using disentangled
latent vectors. Specifically, causality is captured by learning the causal
graph on latent variables through a structural causal model, while correlation
is learned via a novel correlation pooling algorithm. Extensive experiments
demonstrate C2VAE's ability to accurately recover true causality and
correlation, as well as its superiority in controllable data generation
compared to baseline models.
