Gacs-Korner Common Information Variational Autoencoder

We propose a notion of common information that allows one to quantify and separate the information that is shared between two random variables from the information that is unique to each. Our notion of common information is a variational relaxation of the G\'acs-K\"orner common information, which we recover as a special case, but is more amenable to optimization and can be approximated empirically using samples from the underlying distribution. We then provide a method to partition and quantify the common and unique information using a simple modification of a traditional variational auto-encoder. Empirically, we demonstrate that our formulation allows us to learn semantically meaningful common and unique factors of variation even on high-dimensional data such as images and videos. Moreover, on datasets where ground-truth latent factors are known, we show that we can accurately quantify the common information between the random variables. Additionally, we show that the auto-encoder that we learn recovers semantically meaningful disentangled factors of variation, even though we do not explicitly optimize for it.