Learning disentangled representations is an important topic in machine learning for a wide range of applications. Disentangled latent variables represent interpretable semantic information and reflect separate factors of variation in data. Although generative models may learn latent representations and generate data samples as well, existing models may ignore the structural information among latent representations. Described in the present disclosure are embodiments to learn disentangled latent structural representations from data using decomposable variational auto-encoders, which simultaneously learn component representations and encode component relationships. Embodiments of a novel structural prior for latent representations are disclosed to capture interactions among different data components. Embodiments are applied to data segmentation and latent relation discovery among different data components. Experiments on several datasets demonstrate the utility of the present model embodiments.