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Flat Metric Minimization with Applications in Generative Modeling
Thomas Möllenhoff · Daniel Cremers

Tue Jun 11 02:40 PM -- 03:00 PM (PDT) @ Hall A

We take the novel perspective to view data not merely as a probability distribution but as a current. Primarily studied in the field of geometric measure theory, k-currents are continuous linear functionals acting on compactly supported smooth differential forms and can be understood as a generalized notion of oriented k-dimensional manifold. By moving from distributions (which are 0-currents) to k-currents, we can explicitly orient the data by attaching a k-dimensional 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 k-currents. In our theoretical contribution we prove that the flat metric between a parametrized current and a reference current is 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.

Author Information

Thomas Möllenhoff (TU Munich)
Daniel Cremers (TU Munich)

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