# f-GANs in an Information Geometric Nutshell

arXiv: https://arxiv.org/abs/1707.04385 · HTML: https://mlsanity.com/paper/1707.04385/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2017.

## Built on by (most cited first, 9 of 9)

- [Generative Trees: Adversarial and Copycat](https://mlsanity.com/paper/2201.11205/) (2022)
- [Clustering above Exponential Families with Tempered Exponential Measures](https://mlsanity.com/paper/2211.02765/) (2022)
- [A Differential Geometric View and Explainability of GNN on Evolving Graphs](https://mlsanity.com/paper/2403.06425/) (2024)
- [Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding](https://mlsanity.com/paper/2209.07481/) (2022)
- [Generative Adversarial Network: Some Analytical Perspectives](https://mlsanity.com/paper/2104.12210/) (2021)
- [Geometry of EM and related iterative algorithms](https://mlsanity.com/paper/2209.01301/) (2022)
- [Beyond a Single Mode: GAN Ensembles for Diverse Medical Data Generation](https://mlsanity.com/paper/2503.24258/) (2025)
- [Deep Generative Sampling in the Dual Divergence Space: A Data-efficient & Interpretative Approach for Generative AI](https://mlsanity.com/paper/2404.07377/) (2024)
- [Seasoning Generative Models for a Generalization Aftertaste](https://mlsanity.com/paper/2603.18817/) (2026)

## Abstract (arXiv)

Nowozin textitet al showed last year how to extend the GAN textitprinciple to all f-divergences. The approach is elegant but falls short of a full description of the supervised game, and says little about the key player, the generator: for example, what does the generator actually converge to if solving the GAN game means convergence in some space of parameters? How does that provide hints on the generator's design and compare to the flourishing but almost exclusively experimental literature on the subject? In this paper, we unveil a broad class of distributions for which such convergence happens --- namely, deformed exponential families, a wide superset of exponential families --- and show tight connections with the three other key GAN parameters: loss, game and architecture. In particular, we show that current deep architectures are able to factorize a very large number of such densities using an especially compact design, hence displaying the power of deep architectures and their concinnity in the f-GAN game. This result holds given a sufficient condition on textitactivation functions --- which turns out to be satisfied by popular choices. The key to our results is a variational generalization of an old theorem that relates the KL divergence between regular exponential families and divergences between their natural parameters. We complete this picture with additional results and experimental insights on how these results may be used to ground further improvements of GAN architectures, via (i) a principled design of the activation functions in the generator and (ii) an explicit integration of proper composite losses' link function in the discriminator.