Flat Metric Minimization with Applications in Generative Modeling
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:46264635, 2019.
Abstract
We take the novel perspective to view data not as a probability distribution but rather as a current. Primarily studied in the field of geometric measure theory, kcurrents are continuous linear functionals acting on compactly supported smooth differential forms and can be understood as a generalized notion of oriented kdimensional manifold. By moving from distributions (which are 0currents) to kcurrents, we can explicitly orient the data by attaching a kdimensional tangent plane to each sample point. Based on the flat metric which is a fundamental distance between currents, we derive FlatGAN, a formulation in the spirit of generative adversarial networks but generalized to kcurrents. In our theoretical contribution we prove that the flat metric between a parametrized current and a reference current is Lipschitz continuous in the parameters. In experiments, we show that the proposed shift to k>0 leads to interpretable and disentangled latent representations which behave equivariantly to the specified oriented tangent planes.
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