# Flat Metric Minimization with Applications in Generative Modeling

arXiv: https://arxiv.org/abs/1905.04730 · HTML: https://mlsanity.com/paper/1905.04730/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 May 2019.

## Built on by (most cited first, 2 of 2)

- [Controllable Data Generation by Deep Learning: A Review](https://mlsanity.com/paper/2207.09542/) (2022)
- [DeepCurrents: Learning Implicit Representations of Shapes with Boundaries](https://mlsanity.com/paper/2111.09383/) (2021)

## Abstract (arXiv)

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.