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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 defined by an optimization problem over a family of functions and recovers the Gács-Körner common information as a special case. Importantly, our notion can be approximated empirically using samples from the underlying data 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. 

We introduce the Redundant Information Neural Estimator (RINE), a method that allows efficient estimation for the component of information about a target variable that is common to a set of sources, known as the “redundant information”. We show that existing definitions of the redundant information can be recast in terms of an optimization over a family of functions. In contrast to previous information decompositions, which can only be evaluated for discrete variables over small alphabets, we show that optimizing over functions enables the approximation of the redundant information for high-dimensional and continuous predictors. We demonstrate this on high-dimensional image classification and motor-neuroscience tasks. 

We introduce the isoperimetric loss as a regularization criterion for learning the map from a visual representation to a semantic embedding, to be used to transfer knowledge to unknown classes in a zero-shot learning setting. We use a pretrained deep neural network model as a visual representation of image data, a Word2Vec embedding of class labels, and linear maps between the visual and semantic embedding spaces. However, the spaces themselves are not linear, and we postulate the sample embedding to be populated by noisy samples near otherwise smooth manifolds. We exploit the graph structure defined by the sample points to regularize the estimates of the manifolds by inferring the graph connectivity using a generalization of the isoperimetric inequalities from Riemannian geometry to graphs. Surprisingly, this regularization alone, paired with the simplest baseline model, outperforms the state-of-the-art among fully automated methods in zeroshot learning benchmarks such as AwA and CUB. This improvement is achieved solely by learning the structure of the underlying spaces by imposing regularity