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Normalizing flows provide an elegant approach to generative modeling that allows for efficient sampling and exact density evaluation of unknown data distributions. However, current techniques have significant limitations in their expressivity when the data distribution is supported on a lowdimensional manifold or has a non-trivial topology. We introduce a novel statistical framework for learning a mixture of local normalizing flows as “chart maps” over the data manifold. Our framework augments the expressivity of recent approaches while preserving the signature property of normalizing flows, that they admit exact density evaluation. We learn a suitable atlas of charts for the data manifold via a vector quantized autoencoder (VQ-AE) and the distributions over them using a conditional flow. We validate experimentally that our probabilistic framework enables existing approaches to better model data distributions over complex manifolds. 

We address the problem of graph regression using graph convolutional neural networks and permutation invariant representation. Many graph neural network algorithms can be abstracted as a series of message passing functions between the nodes, ultimately producing a set of latent features for each node. Processing these latent features to produce a single estimate over the entire graph is dependent on how the nodes are ordered in the graph’s representation. We propose a permutation invariant mapping that produces graph representations that are invariant to any ordering of the nodes. This mapping can serve as a pivotal piece in leveraging graph convolutional networks for graph classification and graph regression problems. We tested out this method and validated our solution on the QM9 dataset.