  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, $k$-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 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.
