Bridging Disentanglement with Independence and Conditional Independence via Mutual Information for Representation Learning

Existing works on disentangled representation learning usually lie on a common assumption: all factors in disentangled representations should be independent. This assumption is about the inner property of disentangled representations, while ignoring their relation with external data. To tackle this problem, we propose another assumption to establish an important relation between data and its disentangled representations via mutual information: the mutual information between each factor of disentangled representations and data should be invariant to other factors. We formulate this assumption into mathematical equations, and theoretically bridge it with independence and conditional independence of factors. Meanwhile, we show that conditional independence is satisfied in encoders of VAEs due to factorized noise in reparameterization. To highlight the importance of our proposed assumption, we show in experiments that violating the assumption leads to dramatic decline of disentanglement. Based on this assumption, we further propose to split the deeper layers in encoder to ensure parameters in these layers are not shared for different factors. The proposed encoder, called Split Encoder, can be applied into models that penalize total correlation, and shows significant improvement in unsupervised learning of disentangled representations and reconstructions.