# The Fourier Discrepancy Function

arXiv: https://arxiv.org/abs/2102.02979 · HTML: https://mlsanity.com/paper/2102.02979/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 44 references that are in the map.
- First posted on arXiv 5 Feb 2021.

## Builds on (most cited first, 10 of 10)

- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [On Loss Functions for Deep Neural Networks in Classification](https://mlsanity.com/paper/1702.05659/) (2017)
- [A Characteristic Function Approach to Deep Implicit Generative Modeling](https://mlsanity.com/paper/1909.07425/) (2019)
- [Computing Kantorovich-Wasserstein Distances on d-dimensional histograms using (d+1)-partite graphs](https://mlsanity.com/paper/1805.07416/) (2018)
- [Sharp Inequalities for f-divergences](https://mlsanity.com/paper/1302.0336/) (2013)
- [The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems](https://mlsanity.com/paper/2005.06530/) (2020)

## Built on by (most cited first, 1 of 1)

- [From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Properties](https://mlsanity.com/paper/2507.11387/) (2025)

## Abstract (arXiv)

In this paper, we propose the Fourier Discrepancy Function, a new discrepancy to compare discrete probability measures. We show that this discrepancy takes into account the geometry of the underlying space. We prove that the Fourier Discrepancy is convex, twice differentiable, and that its gradient has an explicit formula. We also provide a compelling statistical interpretation. Finally, we study the lower and upper tight bounds for the Fourier Discrepancy in terms of the Total Variation distance.